The probabilities of A, B and C of solving a problem are $\frac{1}{6}, \frac{1}{5}$ and $\frac{1}{3}$ respectively. What is the probability that the problem is solved?A. $\frac{4}{9}$B. $\frac{5}{9}$C. $\frac{1}{3}$D. None of these

Home » The probabilities of A, B and C of solving a problem are and respectively. What is the probability that the problem is solved?A. B. C. D. None of these

The probabilities of A, B and C of solving a problem are $\frac{1}{6}, \frac{1}{5}$ and $\frac{1}{3}$ respectively. What is the probability that the problem is solved?
A. $\frac{4}{9}$
B. $\frac{5}{9}$
C. $\frac{1}{3}$
D. None of these

The probabilities of A, B and C of solving a problem are $\frac{1}{6}, \frac{1}{5}$ and $\frac{1}{3}$ respectively. What is the probability that the problem is solved?
A. $\frac{4}{9}$
B. $\frac{5}{9}$
C. $\frac{1}{3}$
D. None of these

The probability that the problem is solved $=P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A \cap B)-P(B \cap C)-P(C \cap A)+3 P(A \cap B \cap$
C)
Considering independent events, $P(A \cap B)=P(A) \cdot P(B)$
$\begin{array}{l} P(B \cap C)=P(B) \cdot P(C), P(C \cap A)=P(C) \cdot P(A) \\ P(A \cap B \cap C)=P(A) \cdot P(B) \cdot P(C) \end{array}$
Thus, $P(A \cup B \cup C)$ is,
$\Rightarrow \frac{1}{6}+\frac{1}{5}+\frac{1}{3}-\frac{1}{30}-\frac{1}{15}-\frac{1}{18}+3\left(\frac{1}{90}\right)=\frac{5}{9}$
Hence, correct option is b.