Answer:
The value is [tex]P(X = 12) = 0.0356 [/tex]
Step-by-step explanation:
From the question we are told that
The number of boys in the class is [tex]b = 12[/tex]
The number of girls in the class is [tex]g = 8[/tex]
The total number of students is is n = 20
The number of weeks is N = 20
The probability of selecting a girl is mathematically represented as
[tex]p = \frac{8}{20}[/tex]
=> [tex]p = 0.4[/tex]
The probability of not selecting a girl is mathematically represented as
[tex]q = 1- \frac{8}{20}[/tex]
=> [tex]q =0.6[/tex]
Generally the selection of students from the class follows a binomial distribution
i.e
[tex]X \~ \ \ \ B(n , p)[/tex]
and the probability distribution function for binomial distribution is
[tex]P(X = x) = ^{N}C_x * p^x * (q)^{n-x}[/tex]
Here C stands for combination hence we are going to be making use of the combination function in our calculators
Generally the probability that there are exactly 12 weeks in which a girl is selected is mathematically represented as
[tex]P(X = 12) = ^{20}C_{12} * 0.4^{12} * (0.6)^{20-12}[/tex]
[tex]P(X = 12) = 125970 * [1.678*10^{-5}]* 0.016796 [/tex]
=> [tex]P(X = 12) = 0.0356 [/tex]
