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 black flush is a 4 - card hand consisting of all black cards. find the number of possible black flushes. c. find the probability of being dealt a black 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 black flushes. c. the probability is . (type an integer or decimal rounded to six decimal places as needed.)

Explanation:

Step1: Calculate total 4 - card hands

We use combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 52$ (total cards) and $r=4$ (cards in hand).
$C(52,4)=\frac{52!}{4!(52 - 4)!}=\frac{52\times51\times50\times49}{4\times3\times2\times1}=270725$

Step2: Calculate number of black - card hands

There are 26 black cards in a deck. Using combination formula with $n = 26$ and $r = 4$.
$C(26,4)=\frac{26!}{4!(26 - 4)!}=\frac{26\times25\times24\times23}{4\times3\times2\times1}=14950$

Step3: Calculate probability of black flush

Probability $P=\frac{\text{Number of black flushes}}{\text{Total number of 4 - card hands}}$.
$P=\frac{14950}{270725}\approx0.055222$

Answer:

a. 270725
b. 14950
c. 0.055222