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 :-
[tex]P(X=x)=^nC_xp^x(1-p)^{n-x}[/tex]
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:
[tex]P(X=7)=^{20}C_7(0.25)^7(1-0.25)^{20-7}[/tex]
[tex]P(X=7)=\dfrac{20!}{7!(20-7)!}(0.25)^7(1-0.25)^{20-7}\ \ [\because\ ^nC_r=\dfrac{n!}{r!(n-r)!}][/tex]
[tex]=\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[/tex]
Hence, the probability that you will draw a heart seven times = 0.112
Thus , the correct answer is C) 0.112 .
