More Gambling Questions Answered

April 18th, 2022
Back More Gambling Questions Answered

It’s been a few months, so I thought I would come back with another edition of my, “Ask Mission,” articles where I answer gambling-related questions found around the internet.

If you like some of my answers here, then feel free to check out my page for other Ask Mission segments, as well as plenty of other news and editorial writings!

The best way to get ahold of me if you have a gambling related question is to either leave a comment on this article, create an account on our sister site and PM, “Mission146,” or find me on Facebook, “Brandon James,” request me as a friend, and shoot me a DM there. I would suggest that the best way is probably either the comment section here or a PM on WoV, though. You’ll get an answer on Facebook sooner or later, but I sometimes go a few days here and there without checking anything.

Is scavenging abandoned tickets, ‘the ultimate advantage play’?

On Wizard of Vegas, Member McSweeney created a thread (likely in jest) suggesting that scavenging abandoned tickets is the ultimate advantage play.

Of course, it’s not a play at all, and depending on the circumstances, I would say it rises to the level of theft.

As with anything else, I would say that an individual situation depends on the circumstances. While you could certainly never go wrong with printing out the ticket (or picking up one that has been printed) and turning it into security, there are certain situations where I think taking a ticket wouldn’t be morally questionable.

For example, if you happen to find a seat at a bartop at a mostly abandoned bar and are playing video poker, then happen to notice some abandoned credits on a machine near yours and you know the person hasn’t been there for a half hour, or more (fill in whatever duration you like), then you can be fairly confident that they aren’t coming back. I’m not saying that you should take the ticket in that scenario, and I wouldn’t these days, but I don’t think it’s morally questionable.

However, what I did point out in that thread, linking to this WoV article, is that the legality of taking the ticket might vary by jurisdiction, so you want to be careful.

In Colorado, for instance, it’s blatantly illegal to take a ticket, chips or cash that didn’t originally belong to you in any casino—and some players have been prosecuted (successfully) for taking amounts as little as a couple of dollars. In some of these cases, it would seem that the taking of the money wasn’t even intentional, but rather, was a matter of just happening to sit down at a machine that already had some credits on it and playing after inserting their own money.

That’s one state in which I’d be hesitant to even gamble, assuming that they’re that aggressive about it. Could you imagine finding yourself detained, then eventually arrested, prosecuted and fined just because you happened to play a machine that started with $2.00 on it, or something?

It’s also interesting in the case of Free Play because, depending on how the players card system of a particular casino works, it’s possible that there is no way for anyone (except an attendant) to clear the credits without playing them off. Generally speaking, and certainly in every casino that I have ever visited, free play that has been added to the machine can only be played off, there’s no way to cash it out…otherwise card runners would have an absolute field day with that!

In Pennsylvania, it would be illegal to take any cash, tickets, or credits (in any amount) that have been left behind, but I’ve never heard of anyone being prosecuted over small amounts. However, and unlike Colorado, there are clear postings somewhere near the entrance of most Pennsylvania casinos that clearly state that the state doesn’t have a, “Finders keepers,” rule.

With that, if it’s something that you’re even going to think about doing, just make sure that you know the laws of the jurisdiction that you’re in. If you find a decent-sized ticket that is pretty clearly abandoned, and you don’t know the laws, either just turn the ticket in or look up the applicable code. A few bucks really isn’t worth the hassle and, if you’re an advantage player, is definitely not worth getting 86’ed (read: banned from the property) even if there wouldn’t be any legal consequences associated with the act.

Does ultimate x change this royal flush draw?

In a thread over on WizardofVegas, which you could find by doing a Google search and putting some of what you’ll see below in quotes, AlanMendelson asks:

Am I wrong again?

There's a video poker group on Facebook. A player posted a photo of a dealt one card wild royal on a 10 play machine. She held the wild royal but asked on Facebook should she have gone for the natural royal?

I seem to be the only one who says she should not go for the natural.

The machine has multipliers ranging from 2x to 7x on the dealt wild royal.

One Facebooker says the player has a 20% chance of hitting one natural royal and I disagreed with that too.

Am I wrong again?

To which I respond:

MIssion146: The probability of hitting a Royal on a four-to-a-royal draw is 1/47, so the inverse of the probability of missing all of them (the probability of hitting AT LEAST one) is:

1 - ((46/47)^10) = 0.19350860594

So, the probability of hitting AT LEAST one rounds down closer to 19% than 20%, but calling it 20% isn't unreasonable.

My logic on whether or not to keep the Royal is as follows: We know that, on most Deuces Wild paytables (maybe all?) I think it's all of them, because the best worst case scenario for the Wild Royal would be it only pays 20-FOR-1, SF Pays 11-FOR-1 and Flush Pays 3-FOR-1 and the Natural cards are KQJ10 (To make Straight Flushes Available)...and even then you would keep the WRF.

In the case of Ultimate X, the WRF and RF multiplier (4x) should be the same, so no multiplier value is added by going for the Natural Royal. In fact, future multiplier value favors keeping the pat hand. Even if the SF is possible, the only card that would do it would be the Natural Nine as another deuce would just give WRF back. The only possible winning hands are:

RF (1)

WRF (3)

SF (1)

Flush (6)

Straight (6)

The only possibilities that would yield more than a 4x multiplier are the Flush (5x) and SF (12x), but then you have thirty cards that miss and would result in no multiplier. In short, no actual analysis on that is really needed to know that the average multiplier for the following hand is better keeping WRF.

With that, you're quite correct (unless I am missing something) that not only is WRF the right hold, but it's even more favored (multiplier value) than it would be if the game were not UX.

For those of you not familiar with Ultimate X, it’s a video poker game by which a player can earn multipliers for later hands based on winning hands on the current play and the rank of the hand that’s won. While there are single-line Ultimate X games, you’ll usually find three-play, five-play and ten-play games out in the Wild.

What makes Ultimate X a uniquely complicated game, at least in terms of Optimal Strategy, is the fact that the correct hold can differ from the game’s and paytables normal strategy based on not only the potential for future multipliers, but also based on the current multipliers on the board. Players can literally have different holds for the same dealt hand, depending on the situation with the current multipliers.

Obviously, that makes Ultimate X a game where most players are going to give up a lot of RTP by making sub-optimal decisions, so is not one that I would necessarily recommend for casual Video Poker play. The variance of the game is also off the charts, but hey, some players like that aspect. For me, if I were playing a high variance negative expectation game, I would want it to be one for which the holding strategy never changes from normal paytable optimal, such as Super Times Pay or Double Super Times Pay. For games like that, you get the variance of hand multipliers, but also get full return with a single optimal strategy for the paytable.

In the hand above, it was actually a pretty easy situation not requiring an in-depth analysis because the base draw would have you hold the Wild Royal anyway, which the Facebook poster reportedly did, but the fact that this question was even asked gives you some idea of how easy it is to play Ultimate X incorrectly. Not only would you normally keep the Wild Royal anyway, but if there was anything that would change that decision it would be the potential for better future multipliers, which is a non-factor, because the guaranteed multipliers of keeping the Wild Royal are going to be better than the potential multipliers from tossing it.

Does dice setting work?

Over on Wizard of Vegas, MarcusClark asked:

Seriously people do believe that they can throw the dice across the entire table, have them hit the backstop and they can set the dice?

I can understand if they allowed shooters to slide the dice across the whole table that’s one thing, but that would be an invalid attempt at a roll.

How does a casino player actually throw the dice across the entire table, hit the backstop and land on the preset number totally blows my mind. Somebody explain to me what the thought process is, that actually works in the players favor.

To which I responded:

Mission146: I don't think that dice setting actually works, or if it does, it would be an incredibly small minority of people with that degree of precise physical control to exert a meaningful enough influence over the dice.

Anyway, the thought process itself is defensible. A professional bowler and I go down to the bowling alley---the professional bowler will be rolling strikes as a matter of routine while I struggle to even string three strikes together in a single game. Even then, not even a professional bowler is going to roll a strike every single frame...except those rare, but occasional, 300 games.

So, the thought process is that it's physically possible for someone well-practiced enough to exert a sufficient degree of influence over the dice such as to alter the natural probabilities in the player's favor. Again, I've never seen any evidence that would lead me to the conclusion that this works reliably (it would have to be demonstrated over a tremendous sample size of rolls), but the theory that it is a physical task that a person can eventually become good at is a reasonable enough starting point.

If the question was, “Do I think any practicing dice setters are currently enjoying success?”, that answer would be that I certainly haven’t heard of any. Unfortunately for would-be dice setters, it would take a significant sample size of rolls for them to prove out that they had a significant advantage and were actually rolling in a way that made the dice slightly less random as to influence what would otherwise be natural probabilities.

I’m pretty comfortable with my answer that I think dice setting, in theory, is possible as long as you could have a person with the requisite amount of physical skill needed. I just don’t think that such a person actually exists. I think the first thing someone would want to do, to test the theory, would be to create some sort of robotic arm throwing in more-or-less casino conditions to determine whether or not it can throw a significantly uniform shot, and if it can, whether or not that actually matters.

I don’t expect the viability of dice setting to ever be proven, just because it seems like people would rather believe in dice setting than actually test and prove it, but perhaps I’ll be proven wrong one day.

I set the dice just for the optic and have been told not to by casinos on a few occasions. Is the fact that casinos have admonished me enough to prove the viability of dice setting? I don’t think so. I think it just proves that people are capable of being superstitious and believing in things without proof on both sides of the table.

Do ‘mathletes,’ consider recreational players, ‘losers’?

KristMitchell felt the need to do some trolling on WoV like this:

so cvcx says i have an edge an edge to make a whoopin 17 bucks an hour under ac rules!!

now i will i get shuffled up on? will they ask me to leave like i heard over at golden nugget?

how long do card counters live for at one property? will i go on living at ac for several years "like one card counter" and then move to vegas?

tell me the shake down already!!!

i have an edge so the mathies can't say im a loser

but what about the pit crew? banned? half shoed? asked to leave now?

To which I responded:

Mission146: You guys really read a lot into what us math-oriented people think, sometimes. The only thing that we care about is presenting information that is mathematically correct. The notion that, "Mathletes," consider recreational players losers or care what other people do with their own money lives largely in other peoples' minds.

WoV is one of the places where many of us gather. That being the case, yes, if someone comes on here and promotes mathematically inaccurate information, then they will often be called out. Alternatively, if someone presents an interesting or fun question, then they will usually get a good answer.

Anyway, I don't consider recreational gamblers, "Losers." Why would some of the stuff I write be for a general gambling audience? I don't consider gambling addicts, "Losers," I feel badly for them and am sorry that they lost control of what would ideally be a fun and occasional pastime.

Anyway, most of us don't consider these other people losers. I don't even think, "Ploppy," came from the math community; I believe it came from the AP community, but I'm kind of a member of both communities, so whatever...and to whatever extent, "Community," can be used. I'd prefer not to be seen as belonging to any community, but what can you do? I used to use the term, but I try to use, "Recreational gambler," now and have for quite some time.

Who the hell am I to judge anyone? I met someone at the casino a few days ago who is an ambulance driver. Do you think that I think he's a, "Loser?" He transports people to the hospital in a timely fashion so that they don't die...I write about gambling and do part-time advantage play...his profession is 10x more socially useful than the stuff I do and you think I think of him as a, "Loser," because he makes the wrong decisions when he plays (very bad paytable) Ultimate X?

My advice to you is not to ascribe thoughts to other people unless they have directly stated those thoughts. You know that these forums and what, "Mathies," supposedly think aren't the whole world, right---especially when most of us don't even think what you said.

Anyway, I don't know what will happen. You will likely be shuffled up on or flat bet, sooner or later, especially if you're successful and most especially if you become known to the property. If your suggestion is that $17/hour isn't good value, then I will only say there are much more profitable things to do in AC if you are well-bankrolled.

Naturally, he found himself banned from the site in pretty short order. What I can always say for the LCB crowd is that you seem to be a nicer audience than the WoV readers sometimes, who often come around to troll and somehow think that we are looking down on people or trying to tell them what to do because we explain and prove our mathematical positions.

Myself, I’m just trying to give everyone could gambling information. Personally, I don’t care what you play or how you play it with your own money, I would just like players to have access to the starting things they would need to know to pursue advantage play, if they choose. More importantly, I want for casual players to understand the odds that are against them, and the optimal way to play, regardless of what game(s) they select.

We also never tell anyone what to do. While it’s true that we have sometimes informed people when they are making bad bets, they are the ones who came to a public forum and shared how they gamble with everyone. If they hadn’t talked about their play, then I would obviously have no idea what their gambling habits are. I think it’s probably some previously banned person, or perhaps less fortunately, someone who may have had a losing day and wanted to take it out on a forum that’s mostly there to help players. Oh well. What can you do?

Is the video poker cheating!?

On WoV, a member named Jikky asks:

For the past five months I have played NSUD video poker a total of 75838 hands during play I received one royal flush and only seven sets of four deuce hands, my bankroll is down by $2306 , my question is, are these results within expectations or should I assume the machine is rigged and run for the exit!

To which I answered:

Mission146: Your results are consistent with random variations in a fair game and your monetary results are consistent with the fact that you are, "Running badly," on the indicated hands. If you had another Royal (which would technically put you slightly better than expectation on Royals) and were running as expected on Deuces, then you would be ahead overall.

My answer was true, but I guess I might have done better to press him on how sure he was that he was only getting half of the expected number of four deuces.

On that particular paytable, you expect to see Four Deuces just more than once in every 5,350 hands. With that, he would expect to see that particular hand just over fourteen times in the number of hands played. I assume that he is playing at the $1.25 bet level, even though he is down by an unusual amount, if so. In any event, a Royal and running as expected on Deuces would indeed put him ahead if what he says is correct.

Another question that I didn’t ask is how confident is he that he’s playing optimal strategy. If he played 75,838 hands at $1.25 each, then he has nearly $95,000 coin-in played and is down roughly 2.4%. 1.8% of the game’s total return comes from Royal Flushes, and he’s running about 75% of a cycle to the bad also in that regard, so these results don’t see ridiculously bad and are doubtlessly within the third standard deviation even if everything else he said is correct.

I know, you expect games with RTP’s in the high-99% range to pan out faster than that, but with variance, that’s simply not the way it works all of the time. The good news is that you’ll sometimes gets the benefits of that kind of variance and run 2.5% over how you’re expected to perform in 75,000+ hands, perhaps even better if you’re running really well on royals!

Multi-carding question

For those of you who aren’t aware, or haven’t played in United States land casinos, the concept of multi-carding is one by which land-based casinos will send players club offers that include Free Play to patrons. The conditions that need to be met to want to multi-card a casino is that the casino will generally base offers on a player’s ADT (that stands for Average Daily Theoretical, which is assumed to be a loss).

With that, how multi-carding will usually work is that an advantage player first figures out that a particular casino is especially generous with Free Play. That can come by way of either testing his own players club card at that casino, or in the alternative, he can hear from other players that the casino in question is particularly generous.

By, ‘Generous,’ what the player is looking for is a casino that awards an amount of free play that is so significant as to dwarf the expected loss on the first day of coin in. In many cases, the advantage player will want to use players club cards belonging to people that haven’t ever played at that casino before, but that’s not always necessary as long as a significant length of time has passed (which can depend on the casino) since the player last played there.

With that, on WoV, a question was posed about the possibility of players getting banned by a Member called LilRedRooster:

okay, I accept that - there is nothing illegal or untoward about it but I wonder this if you are a highly skilled player won't the casino's computers catch the fact - since you stuck the card in the machine - that they have paid out in cash and comps much more than you have put in? and won't they then take countermeasures to plug the leak?

With that, I responded:

Mission146: The problem with talking about these things at all (DarkOz) is that these sorts of questions are bound to be asked, and in order to successfully defend your claims, you then have to answer in further detail.

Quite simply, there is no evidence necessarily even being given that a person is a, "Highly skilled player."

When it comes to multi-carding, the question is simply one of whether or not the anticipated offers that will come in (i.e. Free Play) is going to exceed the expected loss on the action.

In many cases, people will do the initial play on these cards entirely on slots for a wide variety of reasons:

1.) Because the casino's marketing department will sometimes (not always) treat slots differently than Video Poker and will send slot players substantially better offers such that the expected final outcome (after collecting the free play) is better with slots.

2.) Because the casino's marketing department will sometimes (not always) treat slots differently than Video Poker to the extent that the offer tiers are the same, but the offers can be reached more efficiently on slots due to the fact that slots will require less coin-in (most likely because the offers are ADT based) to reach certain tiers of offers.

3.) Because slots can occasionally be played in +EV situations, or situations in which the perceived EV of the slot play is good enough (such as must-hits, other Progressives, games with progressive Free Games---such as Buffalo Diamond---or games such as the one with the kitty cats) that playing the slots is seen as viable, if not the best thing to do.***

In any event, in many of these cases, there is nothing going on such that the casino would be able to, "Identify a skilled player," prior to extending to the player in question the Free Play offers.

Cash and Comp Value

When someone who intends to multi-card a particular location is deciding whether or not to do that there, this is something that gets, "Tested."

The first thing that someone who intends to multi-card will want is to know that the free play offers, specifically, are going to significantly exceed the expected loss on, "Running the card," which refers to the initial day of play, whether that be slots or Video Poker.

Of course, you can sometimes get a general idea of this prior to even testing a card yourself. If you know (or can speak to) anyone who has played in that casino, get an idea of what they play and what their ADT might be, then that could give you a general idea of whether a casino's free play offers are, 'Good,' or, "Bad.'

Now, you might think that with so many casinos being corporate-owned that this would be pretty easily known depending on the corporation---but that's not true. When it comes to Free Play, individual casinos owned by companies such as Penn National Gaming and CET pretty much have their own marketing departments for direct offers (despite shared national systems for rooms and such, sometimes) and can pretty much market to their own players however they wish.

Your final question referring to, "Countermeasures," also varies from casino to casino and is something that gets, "Tested." Many casinos will, "Kill the card," which either means canceling all offers or requiring the PIN to be reset without killing offers (which usually means offers are going to eventually get killed) if people come in and pick up free play without giving any additional play.

However, even for those casinos that require additional play when, "Picking up," free play, the amount of additional coin-in required just to not have offers already extended be canceled is often pretty low such that the overall expected value of the entire proposition remains very positive.

One of the casinos that I used to multi-card, for example, had a system by which a player only had to play about $2,000 first day coin-in on a Video Poker game (since downgraded) that the casino might not have even known they had. In exchange, at that time, a player would receive almost the same amount back in Free Play over the next six months. Obviously, this well-exceeded the expected loss on the initial 2k coin-in. In fact, you could have blindly picked a slot machine anywhere on the floor and the Free Play you would get back would well more than exceed the expected loss.

Anyway, my experience with that casino was that you just wanted to make sure to earn one point, which was only $5 coin-in, every time that you picked up Free Play on a given card...and I'm not even sure that was absolutely necessary. I only had one card, "Get killed," and then I started to make sure to earn one full point anytime I picked up Free Play and I never had a card, "Get killed," again after that----but the one that was killed may have been killed for some other reason that I'm not aware of.

Thus, the short answer is that everything about the operation varies from casino to casino and the specifics are usually determined by, "Testing," usually multiple cards (play levels) are tested to find the best points.

Some casinos do not bear fruit, usually because the Free Play totally sucks---or that it doesn't totally suck, but that the EV is not such to be worth the time. That said, the casinos that do bear fruit (typically located in highly competitive markets, but sometimes markets with not much competition) more than make up for the ones that don't.

***If you see someone playing a known machine of these types lower than you would be inclined to play it, there is often a reasonably good chance they are playing it before it is +EV on its own to run coin-in on a card.

With that, you pretty much have the basics of multi-carding. It seems that casinos are increasingly cracking down on it, though, and it might eventually reach a point where it becomes illegal. I guess there are some jurisdictions in which you could make an argument that it is illegal, and I know there have been a few arrests regarding it, but I don’t know if those arrests had only to do with the multi-carding.

Obviously, if you’re doing anything where you have someone working for you on the inside, or are perhaps presenting fake ID’s, that’s going to be a huge problem. When it comes to multi-carding, it will almost always be a violation of the casino’s terms and conditions for the players club program, but should not result in an arrest, in most cases, provided that you have the permission of the person whose name is on the card to be using it.

Even then, I would recommend exercising caution if you intend to do this.

There was a follow-up question as to what happens when a taxable jackpot, which in the U.S. means a jackpot of $1,200 or more, is hit using someone else’s card. I can’t answer for how other people handle that, but in my own multi-carding endeavors, when generating the offers, I played games in which hitting a taxable jackpot was either impossible or extremely unlikely. When picking up free play that I had already generated, I would only play Free Play on games in which hitting a handpay is not possible.

Determining keno expected value

BeGoodJohnny44 wanted to know how to calculate the RTP for a Video Keno game, later PM’ing me the paytable as it was not included in his original post. The first thing that I did was solve it, but then I added a disclaimer in case the 9/9 result didn’t multiply, which it didn’t.

How do i determine the house edge of this game . the base game has a house edge of 44.47 but the pay table with the last ball multiplier by four the edge comes out as -111.23. What's my next step to figure this out?

Mission146: Okay, the OP was kind enough to PM me the paytable and says that he must be out of posts for the time being.

Here is the apparent paytable for the nine-spot:

Numbers Selected 9 . In order to win the 2000 dollar jackpot you have to hit 9 out of 9 on a 0.04 cent bet.


4 out of 4 pays 0.08

5 out of 5 pays 0.20

6 out of 6 pays 0.60

7 out of 7 pays 2.40

8 out of 8 pays 8.00

9 out of 9 pays 2000.00

The first thing that we are going to do is simplify this by indexing it to credits won: credit bet, thus:

4/9: 2

5/9: 5

6/9: 15

7/9: 60

8/9: 200

9/9: 50000

Okay, with that out of the way, we have two different events that we need to calculate. The first event is every hit where the last ball is a match multiplies these winnings by four, so that would give you:


4/9: 8

5/9: 20

6/9: 60

7/9: 240

8/9: 800

9/9: 200000

And...Wizard already has a calculator for this:

The return comes out to: 0.985626587407409

However, we can verify that pretty easily. The first thing that we are going to do is look at the conditions for winning, which are simply to match four, or more, numbers. In the case of the nine-spot game, we would need to be starting with at least three of the first nineteen numbers for the 20th number to matter. How many numbers we catch in draws 1-19 will also impact how many possible 20th numbers there are, so with that, let's do some combinatorics:


Match 4:9 + Last Ball:

nCr(9,3)*nCr(71,16)/nCr(80,19) = 0.2320138702607924

Okay, so if this matches the fourth number, then it pays eight units:

0.2320138702607924 * 6/61 * 8 = 0.18256829135

Match 4:9 w/o Last Ball:

nCr(9,4)*nCr(71,15)/nCr(80,19) = 0.0994345158260539

0.0994345158260539 * 2 * 56/61 = 0.1825682913527547016

Match 5/9 with Last Ball:

0.0994345158260539 * 20 * 5/61 = 0.1630074029935309836

Match 5/9 w/o Last Ball:

nCr(9,5)*nCr(71,14)/nCr(80,19) = 0.0261669778489615

0.0261669778489615 * 5 * 57/61 = 0.1222555522451479918

Match 6/9 with Last Ball:

0.0261669778489615 * 60 * 4/61 = 0.10295204399

Match 6/9 w/o Last Ball:

nCr(9,6)*nCr(71,13)/nCr(80,19) = 0.0042107780446605

0.0042107780446605 * 15 * 58/61 = 0.0600553589976169672

Match 7/9 with Last Ball:

0.0042107780446605 * 240 * 3/61 = 0.04970098675

Match 7/9 w/o Last Ball:

nCr(9,7)*nCr(71,12)/nCr(80,19) = 0.0003976279509486

0.0003976279509486 * 60 * 59/61 = 0.02307545813

Match 8/9 with Last Ball:

0.0003976279509486 * 800 * 2/61 = 0.0104295855986518033

Match 8/9 w/o Last Ball:

nCr(9,8)*nCr(71,11)/nCr(80,19) = 0.0000198813975474

0.0000198813975474 * 200 * 60/61 = 0.00391109459

Match 9/9 with Last Ball:

0.0000198813975474 * 200000 * 1/61 = 0.06518490999

Match 9/9 w/o Last Ball:

nCr(9,9)*nCr(71,10)/nCr(80,19) = 0.0000003983522277

0.0000003983522277 * 50000 = 0.019917611385

Okay, so now we will take all of our returns in bold:

Match 4 w/Last: 0.18256829135

Match 4 w/o Last: 0.1825682913527547016

Match 5 w/Last: 0.1630074029935309836

Match 5 w/o Last: 0.1222555522451479918

Match 6 w/Last: 0.10295204399

Match 6 w/o Last: 0.0600553589976169672

Match 7 w/ Last: 0.04970098675

Match 7 w/o Last: 0.02307545813

Match 8 w/Last: 0.0104295855986518033

Match 8 w/o Last: 0.00391109459

Match 9 w/Last: 0.06518490999

Match 9 w/o Last: 0.019917611385

And, we add these all together:

0.019917611385+0.06518490999+0.00391109459+0.0104295855986518033+0.02307545813+0.04970098675+0.0600553589976169672+0.10295204399+0.1222555522451479918+0.1630074029935309836+0.1825682913527547016+0.18256829135 = 0.9856265873727024475

As we can see, this agrees with Wizard's calculator with differences due to rounding, so you can just use the calculator for any other paytables on this game. However, you may have somewhat similar Keno problems to analyze one day, so hopefully this method will be helpful, if so.

One thing that the OP will want to look into is that I know, for games like these, sometimes the top number hit result (9/9 in this case) is the most that it possibly pays, so it doesn't matter whether or not the last number is a hit on 9/9 results as it would not multiply anyway. I would encourage the OP to look into that as it will knock a few percentage points off of the return if $2,000.00 is the highest possible pay and cannot be multiplied.


Match 9/9 with Last Ball:

0.0000198813975474 * 50000 * 1/61 = 0.01629622749

Which then changes the return percentage by:

0.06518490999 - 0.01629622749 = 0.0488886825

Thereby making the return:

0.9856265873727024475 - 0.0488886825 = 0.93673790487

Which seems much more likely to me. I'd suggest that the OP read the Game Rules to determine if that is the case.

ADDED: The OP has informed me that 9/9 does not multiply, therefore, the Return to Player is roughly: 0.93673790487

Would it be bragging to say that one was pretty easy? I thought it was easy, but I have done a TON of Keno math over these many years. I think that I have been at this, to a greater or lesser extent, for nearly a decade at this point. I remember being young and not really knowing much of anything, and now I am probably technically middle-aged, and know even less than I thought I did then, but at least I’m better at math.

Anyway, knowing how to do Keno Math for yourself is also a good way to test online calculators for accuracy. You never know when there might be a bug; it happens sometimes. Besides that, Keno games might eventually come with a stipulation, or perhaps some sort of Free Games feature, that there’s not already a calculator for, so you’ll want to know how to handle that situation.

Either way, I hope sharing this relatively easy way to determine Video Keno (or Live Keno returns) proves useful to someone out there.

To Hedge Or Not To Hedge, For Is It Nobler…

On WoV, DW inquired:

When I signed up for DraftKings’ online casino, one of the bonuses they gave me was a $100 free futures bet to pick the winner of the Super Bowl.

I chose the Rams, at +1200.

Now, of course, I’d like to hedge and maximize my payout without worrying about who actually wins the game, so I came here to ask the experts. What would be the best way to do so, using whatever combination of new-user signup promos and wagers that are possible across the various apps? I’m fortunate that they’re all available to me, since DK was the only casino app I used.

Signing up for FanDuel and taking their “$5 wins $280” promo on the Bengals seems like a no-brainer start.

I live in Michigan, if it helps.

Mission146: Okay, so this is an easy one.

They have Firekeepers Online Sportsbook in Michigan and their current promotion for sports is a $500 wager match. Anyway, you're going to ideally want to deposit $500 and make a straight up $500 bet on the Bengals which is currently at +170 for the line.

The reason that we are doing this with the Firekeepers First Bet Match promotion is because you are going to get the $500, "Risk-Free," bet win or lose. This setup is also going to reduce your variance significantly.

What's going to happen if the Rams win is that you will have $1200 - $500 (Losing Bengals Bet) = +$700 and you will also have the $500 "Free Bet," at Firekeepers.

In the event that the Bengals win, then that profit will be:

500 + (500 * 170/100) = $1350 ($850 of this is profit) AND you will also have the Firekeepers Free Bet.

When it comes to the Free Bet, you can either offset it or not. You'll get more value not offsetting it, I guess. If you want to offset it, however, then simply find a half point line at any sport and make the Free Bet on one of the teams or Over/Under...when you have done that, use a DIFFERENT online sportsbook to make a $250 cash bet on the opposite side of the half point line, which will be really easy if you win on DraftKings as you will have $1200 on DK and a $500 Free Bet at Firekeepers.

So, with the Free Bet, let's say UNLV plays Duke and the Totals is Over/Under 150.5 (Is that reasonable for basketball?)

Anyway, you will bet UNDER 150.5 with your $500 Free Bet and then make a $250 cash bet at a different sportsbook on OVER 150.5. Here are the possible results assuming both lines are -110:

500 * (100/110) = $454.54 - $250 (losing cash bet) = $204.54 Additional Cash Profit

250 * (100/110) = $227.27 (Additional Cash Profit, Free Bet Loses)

If you're really nitty, then you can make the Cash Bet $240 to make the possibilities +$218.18 or +$214.54 if you want them tighter.

Okay, so let's remember how much we make in profits for each event. we have the college hoops game above, so:

Bengals win @ +170 on $500 Cash Bet = $850 cash profit.

Rams Win Futures @ +1200 + losing $500 cash bet = $700 profit.

Again, let's go with the $240 Cash Bet made in opposition to the free $500 bet for the second step, which basically has a return of what we will call $215 as an estimation. That means that your profits will either be about $1065 or $915 depending on who wins the Super Bowl.

***NOTE*** At Firekeepers, it appears that you can do BOTH the casino deposit match as well as this sports bet match, but I am sure there are time limitations, so make sure you read the full terms and conditions if you want to do both promotions. I put the +EV of the Firekeepers Casino Promotion at about +$480 expectation (assumes the terms haven't changed since last I looked), so you can figure out why that is or PM me for more information on that.

If he ended up hedging it, turns out that’s a shame, because the Rams won. On the other hand, Wizard tends to generally advise against hedging on the grounds that it requires making an additional -EV bet on top of the one that you have already made.

The availability of certain promotions at online sportsbooks, however, potentially enable players to make a positive EV (Expected Value) bet and use it as a hedge bet on an earlier bet, as was the case in this situation. My assumption would be that Wizard would take a more charitable view on hedging if an individual is hedging by way of a bet (or promotional setup) that yields a positive expected return on its own. In other words, if betting on the Bengals that way had an ultimate expected value of more than 100% RTP (because of the promotion) in the first place, then what harm could it be in doing that in conjunction with another bet to lock up profits?

I’m also aware of at least one professional sports bettor who, amongst other things, likes to find value in futures bets on teams that are longshots, but he considers likely to get into the Playoffs (or equivalent), at which point, he has the opportunity to hedge by way of picking the team(s) going up against them because he never actually thought they’d win it all in the first place. However, even if the team in question did win it all, he would still come out of it with a net profit.

I won’t say much more than that about him, however, as I am not aware of all of the specifics.

With that, let’s move on to some Facebook questions:

Kathy on Facebook DM’s me:

Brandon, I’ve seen you mention something called the, “Law of Large Numbers,” and was wondering what you mean by that. Can you explain? You seem to think the results come out as expected in the end because of it?

It’s not quite that simple.

Basically, when I refer to the Law of Large Numbers, what I am saying is that results have a tendency to drift back towards expectation over larger and larger sample sizes.

The best way I can think to compare it is to use something like a coin toss. If I toss a coin it’s either going to be heads or tails, right? We will assume that it’s a perfectly balanced coin and a totally fair toss, so what ends up happening is that we end up with either heads or tails. Each of those two things is 50% likely, so is expected to come up 0.5 times in a single toss.

Of course, we know that it will deviate from this expectation, because it has to. The result of the toss will either be heads or tails, so as a result, there will be one of one and zero of the other. Technically, if it lands heads, heads ran twice expectation and tails is running at zero percent of expectation, even though only one or the other could possibly happen.

The Law of Large Numbers basically suggests that, in the long run, this will balance out close to 50% for each side. In other words, that it will eventually get close to the expected value. Casinos basically count on it! However, that’s also why casinos have table maximums, because they need a number of trials such that it’s going to prove out…especially on games with more variance than a coin toss.

Imagine that we toss the coin 100 times and all of those times are heads. I know, that already sounds ridiculous, right? It is pretty ridiculous, with a probability (expressed as a decimal, not a percentage) of 0.00000000000000000000000000000078886091. The odds against such an event coming to pass are 1267650600000000000000000000000 to 1. That’s the same exact thing as doing (2)^100 because that’s how many probabilities that you are selecting from.

However, one interesting fact that you might not be aware of is such is the probability for every specific possible series of 100 coin flips!

Let me clarify: I’m not saying that is the probability of getting something like 47 Tails and 53 Heads—just doing that is FAR more likely. What I mean is let’s say that you flipped a coin 100 times and got this specific series of results, starting from the first flip:

I used the Random Number Generator at Random dot org and got these results where 1 can be heads and 2 can be tails:

1 2 1 1 1

1 1 2 1 1

1 1 2 2 2

1 2 1 2 2

1 2 2 2 2

1 2 2 2 2

1 1 1 1 2

2 1 1 2 2

1 1 2 1 1

2 2 1 1 1

1 2 2 2 1

2 1 2 2 2

2 1 1 2 2

1 2 2 1 2

2 1 2 1 2

2 2 1 2 2

1 2 1 2 1

1 2 1 2 2

1 1 2 1 1

1 1 1 2 1

In any event, that represents 100 coin flips, so the probability that you would have sat down and picked the order in which all of these would have been flipped correctly is the same as all of them being heads! If you had picked heads for the first flip, then you would have had a 50% chance of being right, heads has a 50% chance of being flipped, so it’s the same thing.

In other words, the really long number to 1 odds expressed above also represents the total number of combinations of 100 flips there can be. Whatever the specific result ends up being, certainly not likely to be all heads, it is no more or less likely than any of the other specific series’ of 100 flips.

One thing that many people get wrong is that they think we are saying that results, “Balance out,” in the long-term and everything runs exactly as expected. We’re not exactly saying any such thing. Again, assuming a perfectly fair coin, Expectation would be 50% Heads and 50% Tails, but even if that came to pass, it would change with the very next flip, wouldn’t it?

With that, we know that it can’t be exactly 50% for very long, not if you keep flipping.

The law of large numbers suggests that you will approach expectation given a large enough sample size. It doesn’t say anything about coming exactly to expectation, though you technically would if there were infinite trials, but there’s also no way to observe, ‘Infinite,’ trials.

Basically, it would be ridiculous, as we saw, to flip a fair coin 100 times and have it come up heads all 100 of those flips, right? However, imagine that we flip the coin 100,000 times and heads is leading by 100 after that, not so strange, right? In fact, looking at this binomial distribution calculator, we find that the probability of heads (or tails) to be leading by 100, or more, flips after 100,000 attempts is 0.264347. If you add both heads and tails together, then it is more than 50% likely that one or the other will lead by 100, or more, flips in that sample size.

“How does that benefit the Law of Large Numbers?”, I’m sure you’re wondering. Well, in the ludicrous scenario where Heads is flipped 100 out of 100 times, that means heads was flipped 100% of the time and would still be that. Even a single tails eventually getting flipped benefits the Law of Large Numbers because, as soon as even one flip comes tails, heads can never be 100% in that sample size again. At that point, the sample has begun to approach 50% relative to where it was. 

Another thing that you will note is that we are asking for 50,100, or more, heads. What we have here is 50,100 is only 50.1%, so while heads leads by 100+, heads doesn’t lead by a huge percentage. If you wanted 60% of the flips to be heads, then that would have to be the result 60,000 of 100,000 flips, and…<0.000001 (Not happening). It’s actually WAY less than .000001, that’s just as low as the calculator goes.

Here’s where it gets interesting: Imagine now that there have been one million flips and you want the difference between heads and tails to be at least 100 flips. Using the binomial distribution calculator again, we get a probability of 0.42074 that there will be 500,100, or more, heads. That’s also the probability for there being 500,100, or more, tails…which means a combined probability of 84%+ that either one side or the other will lead by at least 100 flips.

However, leading by 100 flips no longer means as much, because now it means that it leads 50.01% to 49.99%, or more.

Also, if you go back and remember our probability of Heads (or Tails) having been flipped at least 50.1% of the time in 100,000 flips was a combined roughly 52.87%, we can see how the situation changes in 1,000,000 flips. In 1,000,000 flips, one side or the other would have to have been flipped 501,000, or more, times to be 50.1%, or more. The probability of that being the case for Heads is 0.02275 (about 2.275%) and the probability for tails is the same making the combined probability about 0.045501 or 4.5501%.

With that, you can actually see that the disparity in the amount of flips that have gone heads or tails is expected to grow as you do more flips. However, even as the disparity in raw trials continues to grow, the disparity in the percentage of the time a particular thing or another has been flipped continues to shrink.

The highest the percentage can be, as I said earlier, is 100%. After a single flip, one or the other will be 100% while the other will be 0%. As soon as the opposite of the first side is flipped, then the first side can never be 100% again in that sample size.

I hope that helps explain the way that the Law of Large Numbers works. The Law of Large Numbers doesn’t suggest that, because Heads and Tails are equally likely, each of them must be flipped exactly half of the time over x number of attempts. That would be ridiculous and, as I suggested, even if you achieved that at some point in a tremendous sample, it might not even stay that way for long.

Instead, the Law of Large Numbers says that the expectation of 50% will be approached. That’s also not something that happens all at once, either. The coin doesn’t say, “Oh, crap, I’m running at 200 more heads than I should be, let’s go tails the next 200 times in a row,” instead what happens is the disparity gets made up simply by running close to expectation for a very long time.

It’s pretty much the same thing with any gambling game, even though you couldn’t really use a binomial distribution calculator to figure those out…”Binomial,” basically means, ‘Of two possible results.’ However, whether it be Video Keno, Blackjack, Video Poker…or, yes, even slots (in the extreme long-term), your overall results relative to the house edge, and even your results on specific outcomes, will eventually approach expectation.

Of course, a 10/10 in Video Keno has odds of about 1 in 8,911,711.18, so you’re going to be waiting for a long-time if you want that to resolve close to expectation. In most samples, even of a significant size, you’re going to expect that one to either be under or significantly over for a very long time. Of course, “0.00,” I guess is never really that far off of percentage expectation for that outcome!

I received another DM on Facebook from someone named Bryan, who asks:

Brandon, I found a Let it Ride game on an online casino that I liked, so I looked at the Wizard of Odds page and am curious about two holds. The page says that holding four to an outside straight with no high cards and holding four to an inside straight with all high cards both have no house edge; can you explain why? Also, which one would you hold in each situation?

Mission146: The first thing that I will do is knock out the example of the four to an outside straight, because that’s really easy. The example hand we will use is:


And, we are going to assume that you don’t have four suited cards, otherwise it would be four to a straight flush, which you would definitely let ride.

Okay, so we have 48 cards remaining in the deck because four of 52 have been removed, correct? There are four 3’s and four 8’s in the deck, and either of those would give us the straight. In other words, eight out of 48 cards would give us the straight and all of the other cards would cause us to lose.

Given that the straight pays 5 TO 1 (5:1, rather than 5-FOR-1, means your original wager is returned), the math works out like this:

(5 * 8/48) - (40/48) = 0 

In other words, it’s a zero expectation outcome. If you wanted to, you could slide whatever amount you are either letting ride or pull back in the above calculation, because it doesn’t make a difference what the amount of the bet is.

The next EV-neutral decision that you have mentioned is the case in which we have four to an inside straight with all high cards, such as, 10-J-Q-A. Of course, hitting any of these high pairs would pay even money, there are also four cards that can make a straight.

With that, we have three each of tens, jacks, queens and aces…for a total of twelve cards in the deck that will pay out at even money. The four kings in the deck, for this example, would pay out at the 5-TO-1 we mentioned earlier. With all of that, the math works out like this:

(12/48) + (5 * 4/48) - (32/48) = 0

Once again, it doesn’t matter how much you are betting, it’s a zero expectation outcome. In other words, it’s a perfectly neutral decision.

What would I do in those situations? Personally, I would do whatever I felt like doing at the time and I think you should do the same!

It’s not often that you have the opportunity to play strategy-based games with optimal strategies in a non-mechanical way, so there’s no reason to have it in your mind, in advance, of what you’re going to do. It’s an EV-neutral decision, so do whatever you want.

Mathematically, I guess I should disclaim that letting it ride is probably the superior decision IF you are going to continue to play after that hand. The reason why I say that is because you are leaving the money out there in an EV-neutral decision, but you’re making a negative expectation bet when you lay out the money for the next hand. Obviously, an expected value of zero is better than a negative one, so technically it’s probably superior just to let your bet ride unless you’ll be quitting after that hand. You’ll be getting better value on that chip now than you will next time you put it out there.

Oh, also, this is one decision that can easily swing if you happen to know any of the cards of the people sitting around you. They usually don’t make people protect their cards too tightly, so if this is a live game and you see one of the cards, just one, that would make your high pair or straight (assuming it’s the only card you see) now you don’t want to make the bet. If you see literally any card that you don’t need in someone else’s hand, now it becomes a positive EV decision to make the bet!

I’ll quickly prove that in both cases. Imagine that I see a nine of hearts, which isn’t a card that we need in either of the above examples. With that, there are now 47 cards in the deck that we don’t know.

(5 * 8/47) - (39/47) = 0.02127659574

(12/47) + (5 * 4/47) - (31/47) = 0.02127659574

As you can see, it becomes a positive expected value, in both cases, of more than 2% of whatever amount you are betting. If you were betting $5, then your expected gain by changing your decision because you saw that nine would be 0.10638297872 or just over 10.5 cents, which is a pretty decent EV on a $5 bet. Give me a Video Poker game where I have an advantage of over 2% and I would play it all day!

On the other hand, if you saw that another player had one of the cards you needed to make your straight, and it was the only card you saw, then what would happen is this:

(5 * 7/47) - (40/47) = -0.10638297872

(12/47) + (4 * 3/47) - (32/47) = -0.17021276595

YIKES! Now the decision becomes either 10%+ or 17%+ to the bad now that you know one of your straight cards is out of play.

Typically, if you see any cards, you’ll see more than one. More often than not, you’ll see all three of the person sitting next to you, or none of them. If you want to see what seeing one straight card of three, zero cards you need of three, or a high card you could use of three would do to your expected value, then you should be able to follow the same method that I used above to accomplish just that!

The next person to message me on Facebook was Becky, who asks:

Brandon, I saw one of your articles about playing online and was wondering how do you determine whether or not something is a, “Good bonus,” based on the playthrough requirements?

Mission146: There’s actually a lot to unpack in that question, Becky, but I’m going to do my best for you.

The first thing that you have to determine is whether or not the playthrough requirement applies to ONLY the bonus funds, or the deposit + bonus. That should be mentioned either in the Terms & Conditions for that specific bonus, in the general Bonus Terms & Conditions, or sometimes just in the Bonus section of the General Terms and Conditions.

If you can’t find it, or the terms and conditions seem unclear, then you can always contact Live Chat (or send an E-Mail) so that you can get clarification. With that, let’s start with an example.

Imagine that a casino offers me a 50% deposit match on $500, so the matched funds would be $250, in that instance. After reading through their Bonus Terms, I discover that the 20x playthrough requirement applies to BOTH the deposit and the bonus funds. In other words, I would have to play through:

(250 + 500) * 20 = 15,000

In most cases, I’m going to avoid this promotion. Even the best online slots usually only have an RTP of 97% or 98% (when you can find the RTP to begin with–which seems to depend on the casino and the game in question), and even a 2% House Edge (which would be a 98% RTP) would be losing as 15000 * .02 = $300. In other words, you would be getting $250 bonus and the playthrough requirements would lock you into an expected loss of -$300 for a net result of -$50.

Of course, I wouldn’t dismiss this promotion outright without looking at some of the other terms. For example, let’s imagine that this casino doesn’t have any game restrictions on the playthrough, other than Craps, which is extremely common. Okay, so if I can find an online Blackjack game with an RTP of 99.6%, and I only have to playthrough $15,000 in total bets, then I end up with:

15,000 * .004 = $60

In other words, the expected loss on the playthrough is only $60 and the casino is GIVING me $250, so I have an overall expected profit of $190.

However, that’s where it’s important to look at those kinds of terms, as well. It works both ways, if you don’t look at the Terms & Conditions, you might miss good plays or walk directly into bad plays.

Imagine if the Blackjack only contributes 10% to Wagering Requirements. Some people have trouble understanding that verbiage, but the easiest way to figure it out is simply to convert the percentage to a decimal and divide it by the amount of the playthrough, like so:

15000/.1 = $150,000

In other words, now if you wanted to play Blackjack, then you would have to play $150,000 in total bets given that it contributes only 10% to the Wagering Requirements.

At that point, even though a slot with a 98% RTP would still expect you to take a net loss of $50 ($300 loss expected on $15,000 playthrough), it would become much better than playing this Blackjack game, which now comes with an expected loss of $600 on the playthrough bringing you to an expected net loss of all of the bonus funds AND $350 of your original $500. You would expect to deposit $500, get a $250 bonus and walk away with only $150.

Unfortunately, these sorts of promotions are becoming more common, even in the state licensed and regulated online casinos in the United States, so you might as well just hunt for promotions, in my opinion, and play wherever the promotion terms are best.

In any event, it’s been going on for awhile now that many online casinos offer so-called, “Promotions,” where the player is expected to lose even if he/she plays the best game factoring in the combination of both playthrough and RTP. It’s unfortunate, but that’s the way of it. That’s why it is becoming increasingly important to read and understand the full terms and conditions, as well as the basic math to figure these things out, so that hopefully you will only play promotions in which you are expected to profit.


That’s going to do it for this segment of, “Ask Mission.” I actually have a ton of other Facebook questions that I’ve answered, but this has already been a little longer than usual, so I’ll save the best of those questions for another time.

Once again, if you have any questions you want to ask, feel free to get ahold of, “Brandon James,” on Facebook, send me a private message to, “Mission146,” on Wizard of Vegas or leave a comment here and I will likely PM you the answer on this site.

Please keep in mind that I would prefer to publish any answers, so if you ask a question that you don’t want to be made public, my answers will probably be less detailed. I doubt if it was anyone from here, but I’ve had a few people on Facebook basically trying to get me to do a complicated mathematical analysis on a game for free without publishing it. I’m not going to do that, but nor do I accept money directly from people for math work. If you want a detailed answer, then I want the right to publish it.

If you have any questions about any of the answers above, then please put them in the comments! See you next time!

“gambling-related questions found around the internet”

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