I'am analyzing this problem.
From an standard deck of 52 cards four cards are dealed:
-One card is for the house -Three cards are for the players
The player wins if his card is greater than the house card.
So for example if we have:
House - 5 of clubs Player 1 - 7 of spades Player 2 - 3 of diamonds player 3 - 5 of hearts
Player 1 wins the house, the house wins player 2 and player 3.
The problem is to calculate the probability to win, being a player or the house.
I made a brute force aproach with a Perl script. What I have made is to generate all posible permutations 52 nPr 4, and all the favorable permutations for the players (permutations are needed for this aproach, so it can be simulated a card dealed for the house and each player)
Analyzing the problem I realized that a good way to visualize the odds are with a matrix.
Please, look at the first column of the table below as a matrix. The numbers in this matrix symbolize the players cards compared to the house card. If the house card is greater than a player card, I put a Zero. If the house card is equal in number to the player card, I put a "2", and if the house card is lower than the player card, a "1" is put.
"Be the house card x,
if x < player 1 card, x < player 2 card, x > player 3 card", then we have this:
"Be the house card x,
if x > player 1 card, x equals player 2 card, x > player 3 card", then we have this:
Now, with this approach I have got all the combinations for each case:
000 1435200 001 420992 002 121056 010 420992 011 420992 012 54912 020 121056 021 54912 022 7488 100 420992 101 420992 102 54912 110 420992 111 1435200 112 121056 120 54912 121 121056 122 7488 200 121056 201 54912 202 7488 210 54912 211 121056 212 7488 220 7488 221 7488 222 312
At this point is possible to calculate the probability of a player to win. For player 1 it is:
Since the first row simbolyze player 1, we pick the rows with first column equal to "1" and we sum their ocurrences:
Next we divide this for the total of ocurrences:
3057600 / 6497400 = 0.47059.
This is the same probability to win for each player (see the matrix, the permutations are repeated).
Im not 100% sure about the reasoning made in this problem, so comments are welcome.
The next problem is that when I want to analyze the problem dealing five cards, one for the house and four for four players, my computer freeze because there are too much calculations involved. I would like to compute the odds for more players.
I have tried to develop a method of more intelligent counting with no sucess. I suspect, that the numbers of permutations above can be generate with a permutation formula that can be generalized.
Thanks for your help.