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Ask the Wizard #286

What is i^i?

Anonymous

I don't want to just tell you the answer without giving you the opportunity to solve it yourself.

First, here is a hint to help. If you don't already know this equation, you're unlikely to solve it.

Otherwise, I admit I'm bad. Just show me the solution.

For discussion about the equation in the hint, please visit my forum at Wizard of Vegas.

Some of the casinos in Cambodia offer a suited tie bet in Dragon Tiger. It pays 50 to 1 if the Dragon and Tiger cards match in both rank and suit. What are the odds on that side bet?

Johnny from Cambodia

Assuming eight decks, the probability of winning is 52*combin(8,2)/combin(52*8,2) = 1,456/86,320 = 1.69%. The house edge is 13.98%.

On a recent Vegas trip I saw the dealer get a 9-card 21. The rules were six decks and the dealer stood on soft 17. What are the odds of that?

aceofspades

The probability of the dealer getting exactly a 9-card 21 under those rules is 1 in 32,178,035. Here is the probability for various numbers of decks and whether dealer hits or stands on soft 17.

Probability of Dealer 9-Card 21

Decks Stand Soft 17 Hit Soft 17
1 1 in 278,315,855 1 in 214,136,889
2 1 in 67,291,581 1 in 41,838,903
4 1 in 38,218,703 1 in 22,756,701
6 1 in 32,178,035 1 in 18,980,158
8 1 in 29,749,421 1 in 17,394,420

Assuming six decks and the dealer stands on soft 17, here is the probability of the dealer getting a 21 (or a blackjack in the case of two cards), according to the total number of cards.

Probability of Dealer 21/BJ
by Number of Cards

Cards Probability
2 1 in 21
3 1 in 19
4 1 in 56
5 1 in 323
6 1 in 3,034
7 1 in 42,947
8 1 in 929,766
9 1 in 32,178,035
10 1 in 1,986,902,340
11 1 in 270,757,634,011
12 1 in 167,538,705,629,468

Not that you asked, but the next table shows the probability of the dealer making any non-busted hand under the same rules by the number of cards.

Probability of Dealer 17-21/BJ
by Number of Cards

Cards Probability
2 1 in 3
3 1 in 4
4 1 in 12
5 1 in 67
6 1 in 622
7 1 in 8,835
8 1 in 193,508
9 1 in 6,782,912
10 1 in 424,460,108
11 1 in 58,597,858,717
12 1 in 36,553,902,750,535


For more discussion about this question, please visit my forum at Wizard of Vegas.

For casino promotions that still use regular tickets in a real drum (not the electronic ones) where you print your tickets at the players desk and put them into the drum -- do you bend / crease your tickets before you put them into the drum? Do you think that the bent ones have a better chance of being picked?

AxiomOfChoice

I hope you're happy. To answer this question, I bought a big roll of tickets at the Office Depot. Then I put 500 of them in a paper bag, half folded in half, at about a 90-degree angle, and the other half unfolded. Then I had six volunteers each draw 40 to 60 of them one at a time, with replacement, as I recorded the results. Here are the results.

Drawing Ticket Experiment

Subject Folded Unfolded Total
1 25 25 50
2 38 22 60
3 25 15 40
4 34 16 50
5 27 23 50
6 26 24 50
Total 175 125 300


So, 58.3% of the tickets drawn were folded!

If it's assumed that folding had no effect, then these results would be 2.89 standard deviations away from expectations. The probability of getting this many folded tickets, or more, assuming folding didn't affect the odds, is 0.19%, or 1 in 514.

I might add the subjects who drew tickets hastily were much more likely to draw folded ones. Those who carefully took their time with each draw were at or near a 50/50 split.

So, my conclusion is definitely to fold them.

For discussion about this question, please visit my forum at Wizard of Vegas.