Some mathematical exercises by
Jeffrey Rosenthal, Department of Statistics, University of Toronto
This exercise computes various probabilities related to the game of poker.
Recall that an ordinary 52card deck of cards consists of 52 cards, of which 13 each
are Clubs, Diamonds, Hearts, and Spades. Recall that an ordinary poker hand consists of 5 cards, chosen uniformly at random from an ordinary 52card deck. Recall that a poker hand is a flush if all 5 cards are of the same suit, i.e. either all Clubs, or all Diamonds, or all Hearts, or all Spades.
1. Compute the probability that a given ordinary poker hand is a flush.
2. In some poker games, the cards are dealt out a few at a time, rather than all at once.
Suppose a player already has three cards, and they are all Clubs. The player will then
be dealt two more cards, chosen uniformly at random from the remainder of the deck.
Compute the probability that they will end up with a flush. How does this probability
compare with your answer to Question 1?.
3. In some poker games (e.g. "Five Card Stud"), a player gets to see some of their
opponents cards (the "up cards"), before the hand is complete. Suppose a game
is such that each player has been dealt three cards, of which two are "up cards".
Each player will later be dealt two more cards, chosen uniformly at random from the
remainder of the deck. Suppose there are five players in the game. Suppose that one
player ("Player #1") has all three cards Clubs.
(a) Suppose further that, of all the up cards of all the other four players, none of them are Clubs. Compute the probability that Player #1 will end up with a flush.
(b) Suppose instead that, of all the up cards of all the other four players, all of them are Clubs. Compute the probability that Player #1 will end up with a flush.
(c) Suppose the situation (because of the betting so far) is such that Player #1 will fold (i.e., drop out of the game) unless they have at least a 3% chance of getting a flush. Compute the smallest number of Clubs among the up cards of all the other four players, such that Player #1 will fold.
(d) What conclusions can be drawn from this question, regarding an actual game of
poker?
4. In some poker games (e.g. "Seven Card Stud"), players get more than five cards, and
they then get to choose which five cards count as their final poker hand. In this case,
1a hand is a flush if at least 5 of its cards are of the same suit. Suppose in a given
game, each player is dealt a total of 7 cards. Compute the probability that a given
player will obtain a flush. For the next two questions, we will generalise the ordinary 52card deck to an ncard deck (where we will then let n ! 1). We will do this in three different ways, as follows.
(Note that in each case, if n = 52, then we have an ordinary 52card deck.)
(I) The deck consists of n cards (where n is a multiple of 4), of which n/4 each are
Clubs, Diamonds, Hearts, and Spades. [For example, perhaps several ordinary
decks have been mixed together.]
(II) The deck consists of n cards (where n is a multiple of 13), of which 13 belong to
each of n/13 different suits.
(III) The deck consists of n cards (where n is at least 44), of which 13 each are Diamonds, Hearts, and Spades, and the remaining n ? 39 are Clubs.
5. For each of decks (I), (II), and (III) as above, compute the probability that a given
poker hand (consisting as usual of 5 cards chosen uniformly at random from the deck)
will be a flush.
6. Compute the limit as n ! 1of each of the three probabilities in the previous question.
7. Suppose for deck (III) as above, with n a multiple of 52, we consider hands consisting
of 5n/52 cards (instead of 5 cards), and say a hand is a flush if all 5n/52 cards are the
same suit. Compute the probability of such a hand being a flush. (You may assume
for simplicity that n > 13×52 5 , so that Club flushes are the only possible flushes.) Then
compute the limit as n ! 1 of this probability.
If you have time, you may also consider the following. For all the remaining questions,
we consider only an ordinary 52card deck. Recall that an ordinary poker hand is a straight if it consists of 5 cards whose face values are in succession. For example: Ace2345, or 34567, or 8910JackQueen, or 10JackQueenKingAce are all straights. (Note that it is not permitted to "go around the corner", so that e.g. QueenKingAce23 is not a straight.)
8. Compute the probability that an ordinary poker hand is a straight.
9. Suppose a player already has three cards, and their face values are 4, 5, and 6, respectively. The player will then be dealt two more cards, chosen uniformly at random
from the remainder of an ordinary 52card deck. Compute the probability that they will end up with a straight.
10. Suppose a player already has three cards, and their face values are 4, 5, and 8, respectively. The player will then be dealt two more cards, again chosen uniformly at
random from the remainder of an ordinary 52card deck. Compute the probability
that they will end up with a straight.
11. Compare the probabilities you have computed in questions 1, 2, 8, 9, and 10 above.
How might these comparisons affect someone playing an actual game of poker?
