A dogs Smile is a set of seven cards with 4 cards of one den

A dog\'s Smile is a set of seven cards with 4 cards of one denomination and 3 cards if another denomination.,

A card game is played with two ordinary decks of 52 cards each, and each player is dealt a hand of 7 cards chosen at random. What is the probability that a player will be dealt a dog\'s smile, given that the first two cards they get are of the same denomination? (See Homework #2, Question 5 for the definition of a dog\'s smile.)

Solution

The first two cards is given.

Now, there are 50C5 ways to choose the other 5 cards.

There are two cases to get a dog\'s smile here:

1. Choosing 2 cards of the same denomination as the first two, and then 3 cards of another denomination.

The next two cards is then forced, so there\'s only one way to do that.

The other 3 cards can be chosen from 12 other ranks, with 4C3 = 4 ways to choose each.

Thus, there are 12*4 = 48 ways to do this case.

2.

Choosing 1 card of the same denomination as the first two, and then 4 cards of another denomination.

The next card can be chosen 2 ways, from the rank of the first two cards.

The other 4 cards can be chosen from 12 other ranks, with 4C4 = 1 ways to choose each.

Thus, there are 2*12*1 = 24 ways to do this case.

Hence, for cases 1 and 2, there are 48+24 = 72 ways.

Thus,

P(dog\'s smile, given first two is a pair) = 72/50C5 = 0.0000339821 [answer]