# Ask the Wizard #157

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%.

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

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.

Joe from Eureka, CA

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.

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.

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.

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.

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.

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.

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.

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.

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.

Elise

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.

Dog

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%!