Question

a hand consists of 4 cards from a well - shuffled deck of 52 cards. a. find the total number of possible 4 - card poker hands. b. a heart flush is a 4 - card hand consisting of all heart cards. find the number of possible heart flushes. c. find the probability of being dealt a heart flush. click the icon to view a description of a standard deck of playing cards. a. there are a total of poker hands. b. there are possible heart flushes. c. the probability is (type an integer or decimal rounded to six decimal places as needed)

Explanation:

Step1: Calculate total 4 - card hands

The number of combinations of choosing $r$ items from $n$ items is given by the formula $C(n,r)=\frac{n!}{r!(n - r)!}$. Here, $n = 52$ and $r=4$. So, $C(52,4)=\frac{52!}{4!(52 - 4)!}=\frac{52\times51\times50\times49}{4\times3\times2\times1}=270725$.

Step2: Calculate heart flushes

There are 13 heart - cards in a deck. We want to choose 4 of them. Using the combination formula with $n = 13$ and $r = 4$, we get $C(13,4)=\frac{13!}{4!(13 - 4)!}=\frac{13\times12\times11\times10}{4\times3\times2\times1}=715$.

Step3: Calculate the probability

The probability $P$ of an event is the number of favorable outcomes divided by the number of total outcomes. So, $P=\frac{C(13,4)}{C(52,4)}=\frac{715}{270725}\approx0.002641$.

Answer:

a. 270725
b. 715
c. 0.002641