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. a. There are a total of poker hands b. There are possible black flushes. c. The probability is (Round to six decimal places as needed.)

Answer 1

a 270725

b 14950

c 0.0552220

Explanation:

Answer 2

A) there are 270.725 total of poker hands

B) there are 14.950 possible black flushes

C) the probability of being dealt a black flush is 0.0552220

Explanation:

Combinations gives the number of ways a subset of r elements can be chosen out of a set of n elements. Let's use the "n choose r" formula:

`nCr=((n!))/((r!(n-r)!))`

A) the total number of combinations of 4 cards chosen from the deck of 52 cards:

n = 52

r = 4

`nCr= (52!)/((4!(52-4)!)) = (52!)/(4!(48!)) =(52*51*50*49*48!)/(1*2*3*4(48!))`

The 48! terms cancel

`nCr=(52*51*50*49)/(1*2*3*4) = (6.497.400)/(24)  =270.725`

B) Number of possible black flushes:

There are 26 black cards (spades and clubs)

n=26

r=4

`nCr=(26!)/(4!(22!))= (26*25*24*23*22!)/(1*2*3*4(22!))  =(358.800)/(24) \\\\nCr= 14.950`

C) Probability of being dealt a black flush

Simply divide the result of B) over A)

`P=(14.950)/(270.725) = 0.0552220`