Question

A five-card poker hand is dealt from a standard deck. What is the probability that it is a “three-of-a-kind” (three cards with matching faces values, while the other two cards have non-matching face values)? Leave your answer in combinatorial form, clearly indicating (in words) what each term in your answer corresponds to.

Answer 1

As there are 52 cards in a deck, there are totally ​ ways to pick 5 cards.

There are 13 different values of cards in a deck so there are 13 ways to pick the face value that occurs three times [(i.e) A to 10 and K,Q,J].

Thus there are ​ ways to choose one value from this 13 different values (number of ways of choosing the face value for the three cards), then there are ​ ways to choose three cards with matching face value from the 4 cards from the deck. ( number of ways of choosing the suits for these three cards). Thus the total number of ways in choosing three cards with matching face values are.

[Say for example if we want to pick the three 9's from the deck of 52 cards. Then there is one way of choosing 9 from 13 different values in the deck and there are 4 ways of choosing the three 9's from the four 9's in the deck. Thus there are totally ​ ways to obtain three 9's from the deck of 52 cards].

As we have chosen 3 cards already, there are remaining 48 ways of choosing the remaining two cards. That is, there are ​ ways of choosing the remaining two cards. Now to be more precise, as we have already chosen a face value of 9 out of 13 cards, there are 12 ways of choosing the face values for the other two cards . And there are ​ ways of choosing the suits for last two cards. Thus the total number of ways in choosing remaining two cards with non-matching face values are ​ ways.

[Say for example, if the remaining two cards we pick are 7 and 8, then there are ​ ways of choosing these two remaining cards. And 4 ways of choosing one 7 from the four 7's in the deck of remaining 48 cards and 4 ways of choosing one 8 from the four 8's in the deck of remaining 48 cards.Thus there are totally ​ ways to obtain two cards with non-matching face values from the deck of remaining 48 cards].

Thus probability of three-of-a-kind is given by,

P[Three of a kind] =

=

   =

   =

   =

Thus probability of three-of-a-kind is ​.