Answer: The probability of getting a prime number exactly five times = 0.1908
Step-by-step explanation:
Prime numbers from 1 to 30 are 2,3,5,7,11, 13, 17, 19, 23, 29.
The probability of getting a prime number p= [tex]\dfrac{10}{30}=\dfrac13[/tex]
Number of trials n = 12
Binomial probability formula:
[tex]P(X=x) = \ ^nC_x p^x(1-p)^{n-x}[/tex]
, where x= number of successes
n= number of trials.
x = Number of successes
p= probability of getting one success.
The probability of getting a prime number exactly five times:
[tex]P(X=5)=\ ^{12}C_5(\frac13)^5(1-\frac13)^{7}[/tex]
[tex]=\frac{12!}{5!7!}(\frac1{243})(\frac{128}{2187})\\\\=0.1908[/tex]
Hence, the probability of getting a prime number exactly five times = 0.1908
