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Ask the Wizard #157

Suppose we have a gambling game. An unbiased coin flipped repeatedly. For each flip, we have to pay 1 rupee. There are two possible outcomes H or T. If the difference between head tossed and tail tossed becomes 3, we will get 8 rupee from the gambler. Should we play the game, and why? How much is our probability of winning? What should be impact on probability of winning when we are getting 7 or 9 rupees?

Utpal from Lucknow

Lets call x the expected number of flips from the starting point.
Lets call y the expected number of remaining flips if one side is one flip in the majority.
Lets call z the expected number of remaining flips if one side is two flips in the majority.

E(x) = 1 + E(y)
E(y) = 1 + 0.5*E(x) + 0.5*E(z)
E(z) = 1 + 0.5*E(y)

It is then easy matrix algebra to see that E(x) = 9, E(y) = 8, and E(z) = 5. So on average it will take 9 flips for the disparity between heads and tails to be 3. So at 8 rupees it is a good bet for the person collecting the one rupee per flip, because he will receive on average 9 rupees, but pay back only 8. The house edge for the gambler is 11.11%. At 9 rupees it is a fair bet, at 7 the house advantage is 22.22%.

People have a misconception that sports betting limits in Vegas are really high, which in my experience only applies to football and not always in that case. Could you tell me the approximate limits in Vegas sports books on the 4 major sports? Thank you for your time.

Betsy from Malibu

I don’t count hockey as a major sport because it gets very little action. I’m told that the Coast casinos have the highest limits. There is no maximum set in stone that I know of but they take large bets on a case by case basis. Here is what I think they would probably take on the average game.

NFL side: $50,000
NFL total: $5000
MLB money line: $10,000
MLB total: $2000
NBA side: $10,000
NBA total: $2,000

My boyfriend and i have been together for a year, he has a child with his ex, but never gets to see his daughter. He’s 7 years older than me and i’m afraid he’s cheating, early on in the relationship he was having a sexual conversation with a girl he use to work with while he was spending the day with me. And now he’s going out everynight and gets drunk recently i found a message to his friend saying he still has feelings for her. He says he only said it as a friend saying he has feeling for her as a friend and later that same night he text another girl calling her sexy. I just don’t what to do anymore, do you think he’s cheating??

Caitlin from Littleton, Colorado

He probably is. However even if he hasn’t consummated the act of cheating what you have evidence of I would consider lighter degrees of cheating. So even only what is one the table I’d walk away.

Normal Craps are not allowed in California. Here many casinos are using cards to act as dice, using A,2,3,4,5,6 to act as the 6 sides of the dice. I would assume by using multiple decks it would alter the odds. (i.e 4 decks = 16 aces, 16 2’s, ect.) Does this favor the house as in blackjack... or does this favor the player? The player could bet at higher or lower numbers based on the half of the cards out of the shoe before a shuffle(assuming a mid shoe shuffle).


You’re correct, dice alone can not determine the outcome in craps. There are various ways of using cards in place of dice and still have the odds exactly the same. One way is to use two separate decks, thus there is no effect of removal. Another way is to have a 7-card deck, featuring the numbers 1 to 6, plus a seventh "double" card. The first card drawn can never be the double card. If it is then it is put back in and the process repeats from the beginning. If the double card is drawn second then it counts as whatever the first number drawn was. Regardless of how the casino does it I have never seen hard evidence of a case where the odds were different than if two dice were used. So I think you are omitting something from the rules.

I like coming and checking on your website and reading your analysis of games and various probabilities. My question is one that I believe can only be answered by someone who is close to the slot machine industry. There is a slot machine that my girlfriend and I like to play. It is a regular three-reel slot machine with either quarter or dollar denominations. The fun part of the machine is if you get the "hamburger lady" in the 3rd reel and it starts a bonus game. The game involves the player pushing a button that starts a digital die moving and stopping on 1-6 and then the player moves along the hamburger and obtains credits as you go along ultimately trying to get to the top for the huge bonus of 5,000 credits. My question then is that is the bonus already decided for the player before the bonus starts or since after every die move the player has to push the button is the bonus amount completely random. There are various points in the bonus game that if you get a specific die roll the bonus will end but can the player avoid these early terminations or is the bonus amount predetermined even before the start of the game. I say that the bonus is predetermined but my girlfriend believes otherwise. Please help with this argument.

Pedram from Claremont

Your girlfriend is right. Nevada Gaming Control Board regulation 14.040.2(b) states...

"For gaming devices that are representative of live gambling games, the mathematical probability of a symbol or other element appearing in a game outcome must be equal to the mathematical probability of that symbol or element occurring in the live gambling game. For other gaming devices, the mathematical probability of a symbol appearing in a position in any game outcome must be constant."

In other words a representation of dice must have the same odds as real dice. Thus there would be no way to predestine an outcome because the player could roll anything during the course of the game.

Dear Wizard, I have read your comments about online blackjack and the fact that the decks are shuffled after each hand. Is there a casino online that has less shuffling or if not, am I forced to play live dealer blackjack which has more decks stacked and at least I can be aware of when shuffling does occur. Cheers.

Ken from Oxford, UK

Perhaps you misunderstood the point I was trying to make in my blackjack appendix 10. The bottom line is that unless you are a card counter shuffling after every hand, as opposed to using a cut card, is a good thing. The house edge is reduced 0.02% in a 6-deck game, to 0.11% in a 1-deck game, by shuffling after every hand.

Just wanted to let you know that while tickets do expire, generally speaking you can still have them cashed in at the cage. Most of the time they won’t say anything to you at all and if they do, they will cash it in with any sort of pushback from the guest.

Trent from Las Vegas

You are not the only one to take me to task over my comments on ticket expiration dates in my Feb 1, 2006 column. To ascertain whether casinos honor expired tickets I plan to do an experiment. My next trip to the Strip I will get tickets from several casinos and deliberately let them expire and then try to redeem them.

Results of experiment.

In your Nov 28, 2002 column you addressed the proper strategy for the game Pearls Before Swine. They also have a sequel called Pearls Before Swine II. How do I beat this version?

"Anonymous" .

I explain in the 11/28/02 column how to play once there are only three rows left. Here is my strategy for four rows. When it is your turn look up the configuration along the left column and play what is on the right column. For example the starting position of 3456 is listed last and shows you should remove 4 pearls from the row with 5, leaving 1346. If the left column says "Lose" there is no way to win if the opponent plays optimal strategy, which the game at Transcience always seems to do.

A pattern to this table seems to be that you should force the opponent to a situation where the sum of the pearls in the smallest and greatest rows equals the sum of the two in the middle. This would include leaving zero in the row with the least number of pearls.

Pearls Before Swine II Strategy

You Have Leave
1111 111
1112 111
1113 111
1114 111
1115 111
1116 111
1122 Lose
1123 1122
1124 1122
1125 1122
1126 1122
1133 Lose
1134 1133
1135 1133
1136 1133
1144 Lose
1145 1144
1146 1144
1155 Lose
1156 1155
1222 1122
1223 1122
1224 1122
1225 1122
1226 1122
1233 123
1234 123
1235 123
1236 123
1244 1144
1245 145
1246 246
1255 1155
1256 Lose
1333 1133
1334 1133
1335 1133
1336 1133
1344 1144
1345 145
1346 Lose
1355 1155
1356 1256
1444 1144
1445 1144
1446 1144
1455 1155
1456 1346
2222 Lose
2223 2222
2224 2222
2225 2222
2226 2222
2233 Lose
2234 2233
2235 2233
2236 2233
2244 Lose
2245 2244
2246 2244
2255 Lose
2256 2255
2333 2233
2334 2233
2335 2233
2336 2233
2344 2244
2345 Lose
2346 1346
2355 2255
2356 2345
2444 2244
2445 2244
2446 2244
2455 2255
2456 2345
3333 Lose
3334 3333
3335 3333
3335 3333
3336 3333
3344 Lose
3345 3344
3346 3344
3355 Lose
3356 3355
3444 3344
3445 3344
3446 3344
3455 3355
3456 1346

Brad S. wrote in to add a general strategy for any number of pearls and rows. First you break down each row into its binary components. For example the starting position of the Transcience game would be as follows.

  • 3 = 2+1
  • 4 = 4
  • 5 = 4+1
  • 6 = 4+2

Then you endeavor to leave an even number of each power of 2. For example in the above there are two 1’s, two 2’s, and three 4’s. So there is an extra 4. You then remove 4 from any of the rows with a 4 term. Keep doing this until you can get your opponent down to 2,2 or an odd number of 1’s.

Try this strategy on the Pearl 3 game, you’ll win every time. If you start with a losing scenario as I did on game 10 (4+7+8+11) you can click on "go" to make him go first.

What are the odds of having pocket aces and pocket kings both being dealt in the same hand?

Jake from Loveland, CO

The odds of a specific player having aces is combin(4,2)/combin(52,2) = 6/1326. The odds of the next player having a pair of kings is combin(4,2)/combin(50,2) = 6/1225. However, in a ten-player game there are 10 possible players who could get the aces, and 9 possible players for the kings. So a strong approximation would be 10*9*(6/1326)*(6/1225) = 0.001995, or 1 in 501. This answer is slightly too high, because it double counts the situation where two players have aces, or two have kings, or both.

Sir! What is or where we can find the best basic strategy for Club Blackjack as played in Finland, Sweden and Baltic Ferries. 6d, NHC, DD9-11,1Spl, NDAS,dealer wins ties 17-20 (17-19 in Swd). We understand that this is a complete suckers game with house edge of nearly 9% but it is the only game in town and since we cannot win we would at least die fighting as good soldiers are expected to do. Thank You!

Vankka from Vantaa, Finland

I’ve actually seen those rules when I went to Helsinki in 1986. I was only 20 at the time and I’m embarrassed to say I actually played the game, because I still underage in the U.S., and was still years away from becoming the Wizard. In fact it would have been the first legal game of blackjack I ever played. Meanwhile there was a single-zero roulette table right next to it.

I assume the dealer stands on soft 17 in both cases. It is my understanding that double after a split is allowed in Sweden but not Finland. Assuming that I get a house edge of 5.99% in Sweden and 8.93% in Finland. Here are the basic strategy charts.

I have this problem with my boyfriend. We started dating the end of January last year, and he broke up with me in the begining of October last year, because he cheated on me. He just called me asking if I would take him back about 3 weeks ago. So i took him back, so were together again. But something still doesn’t feel right. I just got this feeling that something is really wrong. I get so sick to my stomach and just have all these thoughts going through my head about him but i just cant figure out what is so wrong. It’s driveing me crazy, I can’t consitrate on anything anymore. Do you have any idea what might be wrong?

Brittany from Wausau

My diagnosis is that your conscience is saying you are making a big mistake taking him back. The sick feeling is what happens when you know you aren’t doing the right thing. So listen to your gut and throw him back.

If I have a bet with another person, which is witnessed by three others and we shake on the bet. Is it by law a legal bet and must be paid?

Thomas from Durham

No. Illegal debts are not enforceable. I’m not a lawyer but as I understand it unless the law specifically allows for gambling on whatever you bet on, and somebody had a license to take the bet, then the bet would have no legal protection. That is why I list "Honor thy gambling debts" as the first of my Ten Commandments of Gambling. A true gentleman honors all his debts, but especially ones based on honor, like a bet made only verbally or with a handshake.

Me and my guy just started a month ago but I’m already having second thoughts about our relationship. You see, he’s two hours apart from me and all we have is messaging over the phone and talking over the phone for the pass one whole month. During our first few weeks together, I believed everything he said, including he’s stop smoking for me and he’s trying to make a change for me. Now, after some small argument, i find it hard to believe every time he said he’s busy and I’m actually not believing that he’s gonna change to be a better person anymore. Last time, when we were dating, a girl messaged him and told him her affection and he just said sorry he’s sleepy now he’s got to go. Now, i’m really worried I’m being treated this way too. We’ve only been together for a month and I find it hard to trust him already. Is it all my mistake? Should i stop this?


I would dump just for the reason of smoking alone. Smoking is a vile, filthy, disgusting, and repugnant habit. Not only is he killing himself, you also put your life at risk by exposing yourself to his secondhand smoke. I know you said this is long distance but it can’t stay that way forever. The most important piece of dating advice I can give is that a non-smoker should never even consider dating a smoker. That is three immediate strikes. Don’t even let them get up to the plate. Furthermore, I wish the state would ban smoking in casinos.

I don’t believe him when he said he would quit for you. He probably has all kinds of action where he lives and is just keeping you on the back burner for when he needs a change of venue. I also don’t see the reason for starting out a long-distance relationship. The odds against these are bad to begin with, but especially with no kind of history together to fall back on. So, yes, it is all your mistake. You deserve a lot better. Next time make it somebody local and a non-smoker.

In a recent Ask the Wizard column, a reader wrote asking about a $2 video BJ game in North Carolina that pays $13 for a BJ and $1 for any other win. What the reader did not understand is that the payouts are NOT 13:2 for BJ and 1:2 for other wins... instead the payouts are 13 for 2 on BJ and 1 for 2 on all other wins. Like a slot or video poker machine, this machine takes your bet as soon as you hit the "Deal" button. Then, if you "win", it pays $13 for BJ or $1 for OW. In other words, the player actually loses money on any non-BJ win. Given these payouts, what is the EV for a single deck game?


And I thought the rules in Finland were bad. I’m assuming that ties lose. It wouldn’t make any sense for ties to push, which would be better than a win. So if ties lose, wins lose half, and blackjacks pay 11 to 2 the house edge would be 51%!