probability

Question

find the probability that a poker hand is 4 of a kind ( 4 cards of the same rank and any other card )

 

Thank you!!!!!

Answer 1

There are 52 cards in a deck - 13 sets of 4 cards sharing the same rank. As each card is dealt, the number of cards in the deck decreases, for which the solution must account. One issue here is that you did not mention how many players are in the game. In the real world, the number of players affects the number of cards from which your cards are dealt. However, I suspect this problem wants you to assume that only one hand is dealt from a deck of 52.

 

The first card is dealt from a full deck of 52, but it can be any card. This is a 100% probability.

 

We want the next 3 cards to have the same rank as the first, so we are looking for 3 particular cards from among the remaining 51 cards. The fraction 3/51 represents the 2nd card. Each time a card of the same rank is dealt, it reduces the number of cards of that rank, as well as the number of cards remaining in the deck. Therefore, 2/50 represents the 3rd card & 1/49 represents the 4th card.

 

As the 5th card can be any other card, this is a 100% probability & does not affect the overall probability. Also, the order these cards are dealt does not matter - you will get an appropriate hand at the correct probability regardless of the order the cards are dealt.

 

To finally compute the probability, we multiply these fractions:

 

3/51 x 2/50 x 1/49

 

You might already realize this is a low probability - you could almost just forget about it. Do the math & see what you get - I got 6/124950 - less than 6 in 124 thousand - which comes out to 0.000048 less than 5/1000 of 1%.