Question

You read in a book about poker that the probability of getting a hand containing "four of a kind" is approximately 0.00024. This means that

a. in a very large number of hands, the proportion of hands containing four of a kind will be approximately 0.00024.
b. in every 100,000 hands, you will receive a hand containing four of a kind exactly 24 times.
c. in a very large number of hands, the percentage of hands containing four of a kind will be approximately 0.00024%.

Answer 1

a

Explanation:

There are two ways of determining the probability of getting “four of a kind” in a poker hand.

The first one is experimental.

We first select an allegedly fair deck of cards. This is a deck in which every single card has the same possibility of being selected, then we deal a large number of poker hands and compute the number of hands dealt and the number of hands in which we got “four of a kind”.

The proportion

(number of hands in which we got “four of a kind”)/( number of hands dealt)

This number would give us an approximate proportion of the number of hands in which we got “four of a kind” in an ideal situation.

The other one is theoretical.

We compute the exact amount of poker hands that can be dealt with an “ideal” fair deck of cards and compute the exact amount of hands in which we got “four of a kind”.

This can be done with techniques of counting which include combinatorial analysis.

Next, we compute the proportion

(exact amount of hands in which we got “four of a kind”)/(exact amount of poker hands that can be dealt)

and we get the theoretical probability.

The experimental result should be very close to the theoretical result if the experiment is repeated a large number of cases.

In any case the correct answer is

a. in a very large number of hands, the proportion of hands containing four of a kind will be approximately 0.00024.