Answer to Question #110622 in Statistics and Probability for Rojan

Question

In a class, there are three sections, each including 20 students. In first section there are 10 boys and 10 girls, in second section, there are 15 boys and 5 girls and in the third section, there are 12 boys and 8 girls. Five students are selected from each group to form a committee of 15 students. What is the probability that all 15 students selected are girls?

Answer 1

Total number of methods to select 5 students= $\binom{20}{5}=\frac{20!}{5!15!}=15504$

In the first section there are 10 boys and 10 girls,

Number of methods to select 5 girls and 0 boys= $\binom{10}{5}*\binom{10}{0}=\frac{10!}{5!5!}*1=252$ ,

probability of selecting 5 girls and 0 boys $(P_1)=\frac{252}{15504}$

In the second section there are 15 boys and 5 girls,

Number of methods to select 5 girls and 0 boys=$\binom{5}{5}*\binom{10}{0}=1*1=1$

probability of selecting 5 girls and 0 boys $(P_2)=\frac{1}{15504}$

In the third section there are 12 boys and 8 girls,

Number of methods to select 5 girls and 0 boys=$\binom{8}{5}*\binom{12}{0}=\frac{8!}{5!3!}*1=56$

probability of selecting 5 girls and 0 boys$(P_3)=\frac{56}{15504}$

probability of all 15 students being girls=$P_1*P_2*P_3$

probability of all 15 students being girls=$\frac{252}{15504}*\frac{1}{15504}*\frac{56}{15504}$

probability of all15 students being girls=$\bold{3.7866*10^{-9}}$