Answer to Question #317730 in Statistics and Probability for Fish

Question

the probability that applicants A, B and C are successful are 1/3, 2/3, 1/4 respectively. if the three were invited what is the probability that only two of them were successful

Answer 1

$P\left( A \right) =\frac{1}{3},P\left( B \right) =\frac{2}{3},P\left( C \right) =\frac{1}{4}\\P\left( only\,\,two\,\,successful \right) =P\left( A \right) P\left( B \right) \left( 1-P\left( C \right) \right) +P\left( A \right) \left( 1-P\left( B \right) \right) P\left( C \right) +\left( 1-P\left( A \right) \right) P\left( B \right) P\left( C \right) =\\=\frac{1}{3}\cdot \frac{2}{3}\cdot \frac{3}{4}+\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{4}+\frac{2}{3}\cdot \frac{2}{3}\cdot \frac{1}{4}=0.305556$