I watch with some mild interest on TV the poker games when I have little better to do at night. And I notice these guys playing Texas Holdem Poker.
My question is, the TV has beside each players name a set of cards, beside this they have percentages, these percentages change as the game evolves. How are these percentages worked out.
If I choose to play online. Given I cant see other peoples cards, and without hacking or cheating in other ways. How close can I get to these percentages if I had software to read only my cards and the up cards on the table as the game progresses. Remember, no cheating or hacking.
Best Answer
The percent showing up in the corner of the screen and the percent in a player's head are actually answers to subtly different questions.
The television percent is answering the question "Given perfect information about all players' cards, what is the likelihood that a particular player will win?" This can be computed by considering all possible board (the face-up cards) outcomes. If 25% of these outcomes give you a winning hand, then you have a 25% chance of winning. There are too many possibilities for a human to consider, but a computer will handle it easily.
The player is answering the question, "Given that I don't know my opponent's hand, what is the likelihood that mine will prevail?" Here, we might try to proceed by considering all possible hands for the opponent and all possible board cards - still reasonable for a computer.
To appreciate the difference between these two questions, consider the following example. You have Kh Qh, your opponent has 4d 5d, and the board is 2h 6h 9h Jh As (so it is $K$-high flush versus high card). The television reports 100% in your favor and 0% for your opponent; all the cards are in play and you hold the winning hand. Since you don't know your opponent's hand, however, you still fear he may hold an Ah - the only hand that will beat you. Purely mathematically, you could estimate the likelihood of him holding the Ah as follows:
There are $\binom{45}{2} = 990$ two-card hands that can be formed from the 45 unknown cards. Of these, 44 contain the Ah (select the Ah, then pair it with any of the other 44 cards). Thus, the chance of his having the Ah is $$ \frac{44}{990} \approx .04. $$
So, while everyone watching the game on television knows you are the winner, you are roughly 96% sure that you are the winner. As you might suspect, the discrepancy between these two answers can be much larger when more cards remain to be placed on the board.
As a more extreme example of how greatly these two calculations can differ, suppose the entire board is in-play with you and your opponent both holding extremely poor hands (but yours is slightly better). The television will again say 100% in your favor, but you will figure your chances of winning at next to zero!