Question

The probability of drawing a heart from a standard deck of cards is 0.25. You record the car do you draw and return the car before shuffling to ensure a binomial distribution. If you do this 20 times, what is the probability that you will draw a heart seven times.
A) 0.888

B) 0.001

C) 0.112

D) 0.473

Answer 1

Answer: C) 0.112

Step-by-step explanation:

In binomial distribution with parameters n (Total trails) and p (probability of getting success sin each trial) , the probability of getting success in x trials is given by :-

$P(X=x)=^nC_xp^x(1-p)^{n-x}$

Given : The probability of drawing a heart from a standard deck of cards is 0.25

Here , getting heart is the success.

Then p= 0.25

For n= 20

The probability that you will draw a heart seven times i.e. x= 7:

$P(X=7)=^{20}C_7(0.25)^7(1-0.25)^{20-7}$

$P(X=7)=\dfrac{20!}{7!(20-7)!}(0.25)^7(1-0.25)^{20-7}\ \ [\because\ ^nC_r=\dfrac{n!}{r!(n-r)!}]$

$=\dfrac{20\times19\times18\times17\times16\times15\times14\times13!}{7!13!}(0.25)^7(0.75)^{13}\\\\=77520(0.00006103515625)(0.0237572640181)\\\\=0.112406195476\approx0.112$

Hence, the probability that you will draw a heart seven times = 0.112

Thus , the correct answer is C) 0.112 .