Question

In Texas Hold'em, 2 cards are first dealt face down (hole cards) to each player, and then five community cards are dealt face up. Each player seeks the best five card poker hand from any combination of the seven cards of the five community cards and their two hole cards.

a) During a self-practice, when only two hole cards are dealt, what is the probability that you get a pair of aces?

b) During a self-practice, assume only seven cards are dealt. What is the probability that you are able to form a “Four of a kind” 3 from the seven cards, given that your two hole cards are both Aces. (make sure you have thought about all possible scenarios.)

c) In an actual Texas Hold’em game, only John and you are left on the table after all the cards are dealt. There are $100 in the pot from the previous bets, $20 of which are from you. John decides to bet another $80. You choose to call with the same amount of money. Hence, there are now $260 in the pot, $100 of which are from you. You take all the pot money if you win, but none will be returned if you lose. Suppose your win rate is 30% and there is no draw. What is the expected profit you get from making this call?

d) Another option you have is to fold and forfeit after John places the bet. This means you will not get anything back from this game, including your previous bet money ($20). Compare to your answer in Part (c), which is the better option for you to make? Why?

Answer 1

a) I need to draw 2 out of the four aces, when choosing from a total of 52 cards

b) There are two scenarios.

Case I - Two out of the 5 face up cards are also aces. We need to select both aces, and then select any 3 from the remaining 48 cards.

Case II - Four out of the five cards are from the same suit. We need to select one out of remaining 12 suits, whose all cards will be selected and the select any one out of the remaining 46 cards.

The required probability is

c) If I win, I shall make the difference of total and my bet money, viz. $160. If I lose, I shall give away $100

There is an expected loss instead of profit, of $22

d) Here the loss is $20 instead of $22 in previous case. So I shall go with this option.