What is the probability of being dealt four cards of a kind in a five card hand?
I know it is 0.00024, but I need this as a fraction and I don't know how to do that.
A four-of-a-kind is four cards showing the same number plus any other card.
If we order the 5-card hand with the four-of-a-kind first, we have C(13,1) choices for the number showing on the first four cards. Since we will have all four suits, we have only C(4,4) = 1 way to choose the suits. The
remaining card will be any of the 48 remaining cards:
# 4-of-a-kinds = C(13,1)*C(4,4)*C(48,1) = 13·1·48 = 624
Number of possible 5 cards from 52 = C(52,5) = 2598960
Dividing by the number of possible hands gives the probability:
P(4 - of - a - kind) = 624/2,598,960 = .000240096
To find the probability as a fraction, we need to divide the number of favorable outcomes by the total number of possible outcomes.
In this case, we want to find the probability of being dealt four cards of a kind in a five card hand. A four of a kind hand consists of four cards of the same rank plus any fifth card.
First, we need to determine how many ways we can have four of a kind. There are 13 different ranks in a deck of cards, so there are 13 possibilities for the rank of the four cards. Once we choose the rank, there are 4 cards of that rank in the deck, so we need to choose 4 cards out of 4, which is 1 way.
Now, we need to determine the possible choices for the fifth card. Since we've already chosen the rank for the four cards, there are 48 remaining cards in the deck that are different from the four cards of the same rank. We need to choose 1 card out of 48, which is another 48 ways.
So, the number of favorable outcomes (ways of getting a four of a kind in a five card hand) is 1 * 48 = 48.
Now, we need to determine the total number of possible outcomes (ways of getting any five card hand) from a deck of 52 cards. First, we choose 5 cards out of 52, which can be calculated as 52 choose 5.
Using the formula for combinations, we know that 52 choose 5 is equal to (52!)/(5!(52-5)!), which simplifies to (52!)/(5!47!).
The factorial function (!) is the product of all positive integers up to a given number. We can calculate factorials using the following steps:
- 52! = 52 * 51 * 50 * 49 * 48 * ... * 1
- 5! = 5 * 4 * 3 * 2 * 1
- 47! = 47 * 46 * 45 * 44 * ... * 1
Now, we can calculate the total number of possible outcomes by substituting the values into the formula. After simplification, it becomes:
(52 * 51 * 50 * 49 * 48 * ... * 6 * 5!)/(5! * 47 * 46 * 45 * ... * 1)
This simplifies to (52 * 51 * 50 * 49 * 48)/(47 * 46 * 45 * ... * 1).
Therefore, the total number of possible outcomes is (52 * 51 * 50 * 49 * 48)/(47 * 46 * 45 * ... * 1).
Finally, we can calculate the probability as a fraction by dividing the number of favorable outcomes (48) by the total number of possible outcomes [(52 * 51 * 50 * 49 * 48)/(47 * 46 * 45 * ... * 1)].
So, the probability of being dealt four cards of a kind in a five card hand is 48/[(52 * 51 * 50 * 49 * 48)/(47 * 46 * 45 * ... * 1)].