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# An Interesting Question about Toking

3 August 2007

Recently one of the readers of Casino City Times, Andrew S., sent Frank a question involving the toking of Texas Hold'Em dealers. Frank sent the question to Bill Burton, who in turn sent it to me. So, Andrew, I guess you're stuck with me. Here is Andrew's letter to Frank:

Frank,

This question may not be for you, but I couldn't find a resident odds person with a listed email address through the emails sent to me. If this question is better answered by someone else in the network, please forward it to them. Thanks!

I just returned from a night of playing poker. To make the already lively 3/6 limit table in a better mood, another player and I made a small proposition to the dealer. After all action was completed pre-flop, the dealer would name three ranks. If two of the three showed up on the flop, I would toke the dealer \$1; if all three \$5. The other player was a little more generous and would toke \$1 if two of the three showed up anywhere on the board (no \$5 for all three, though).

I know it's a losing proposition for us as there is no return. However, what is the dealer's expected return on this "free" proposition?

With my proposition, I calculated the chance of getting all three on the flop at 0.289%, or roughly one in every 350 hands. I also calculated the chance of two on the flop at 11.1%, or about one every 9 hands.

With the other player's proposition, I calculated the chance of getting two of the cards on the board at 30.6% or roughly three out of ten hands.

Assuming 20 hands an hour, the dealer should expect to hit two on the flop twice (\$2/hour), but only all three after about 17 hours of dealing (\$0.29/hour); he should also expect to hit two on the board six times (\$6/hour). This totals to roughly \$8.75 and hour.

Is my math correct or did I miss something somewhere? If so, I think it's a great game for the dealer to play. He is getting free money for picking three random cards - how can being toked be any easier?

Thanks for you time!

Andrew S.

The rules were not quite clear to me so I wrote to Andrew for some clarification. Having done this let me state the rules precisely. If on the flop at least two cards of the three ranks chosen show up, the dealer is toked \$1. This includes two or three cards of the same rank. If three cards of three different ranks show up the dealer is toked \$5.

To analyze this proposition I will use the notion of a combination and use the notation C(k, n) to represent the number of combinations of n things taken k at a time. I have discussed this idea before on this web site. FYI, the formula is

C(k, n) = n! / [(n - k)!k!] where n! = n(n - 1)(n -2) ... 3 x 2 x 1

To begin with, this proposition must be calculated using a full 52-card deck. You may think this strange since there are less than 52 cards in the deck when the dealer deals the flop. However since the dealer has seen none of the table cards at this point, the fact that the cards are on the table rather than in his deck is irrelevant. Here is another way to look at it.

Suppose we play the game a bit differently. The dealer deals out the three-card flop at the start of the game but doesn't turn the cards over. Then play continues as usual. The flop is exposed only after all of the initial two-card play has been completed. This game is equivalent to the usual Hold'Em game but it is clear that the flop has been dealt from a 52-card deck.

Next, note that the order of the cards in the flop is unimportant. That is why the analysis uses combinations rather than permutations. This remark and the remarks above apply to the second player's proposition as well.

How many ways can we get three of the dealer's choices on the flop? Well, there are twelve cards to choose from so there are C(3, 12) or 220 possibilities. However, 4 x 4 x 4 or 64 of these are flops with no matches. Thus there are 220 - 64 or 156 \$1 tokes when three appear.

For exactly two there are C(2, 12) or 66 ways of choosing two of the twelve and C(1, 40) or 40 ways of choosing the other card; the product is 2640. For exactly one we have C(1, 12) x C(2, 40) or 9360 and for exactly none we have C(0, 12) x C(3, 40) or 9880. If you're wondering why I calculated these last two it is to check my calculations. All of my numbers, if correct, must add up to C(3, 52) or 22,100, which is the number of possible three-card flops. The results are tabulated below.

 Flop Type Frequency Toke Product 3 Matched 64 5 320 3 Unmatched 156 1 156 2 Dealer's 2,640 1 2,640 1 Dealer's 9,360 0 0 No Dealer's 9,880 0 0 Totals - 22,100 -- 3,116

The expected return to the dealer is 3,116 divided by 22,100 and is approximately 0.141.

The calculations for the other player are similar to those above except that 5 cards are used. Here is my tabulation of the results:

 Flop Type Frequency Toke Product 5 Dealer's 792 1 792 4 Dealer's 19,800 1 19,800 3 Dealer's 171,600 1 171,600 2 Dealer's 652,080 1 652,080 1 Dealer's 1,096,680 0 0 No Dealer's 658,008 0 0 Totals - 2,598,960 -- 844,272

The expected return to the dealer is 844,272 divided by 2,598,960 and is approximately 0.325.

Altogether the two propositions have an expected return of 0.325 + 0.141 or 0.466 per game. Assuming 20 games per hour this results in an expected return to the dealer of approximately \$9.32 per hour.

So, Andrew, I hope this clears up any questions you have. See you next month.

Don Catlin can be reached at 711cat@comcast.net

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Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers