Question

Let $P_1,P_2$ and $P_3$ are the probabilities of a student passing three independent exams $A,B$ and $C$ respectively. If $P_1,P_2$ and $P_3$ are the roots of equation $20x^3-27x^2+14x-2=0$ , then the probability that the student passes in exactly one of $A,B$ and $C$ is

• A. $\frac{3}{20}$
• B. $\frac{7}{20}$
• C. $\frac{1}{4}$
• D. $\frac{1}{5}$

Answer 1

Answer: (C)

$P_1+P_2+P_3=\frac{27}{20}$
$P_1{P}_2+P_2{P}_3+P_3{P}_1=\frac{14}{20}$
$P_1{P}_2{P}_3=\frac{2}{20}$
Probability that the student passes in exactly one of $A,B,C$ is
$=P_1+P_2+P_3-2\left(P_1{P}_2+P_2{P}_3+P_3{P}_1\right)+3P_1{P}_2{P}_3$
$=\frac{27}{20}-2\left(\frac{14}{20}\right)+\frac{6}{20}=\frac{5}{20}=\frac{1}{4}$