Assertion (A) : The probability that candidates A and B can solve… - Vidyadip

MCQ

Assertion $(A)$ : The probability that candidates $A$ and $B$ can solve the problem is $\frac{1}{5}$ and $\frac{2}{5},$ then probability that problem will be solved is given by $\frac{12}{25}$.
Reason $(R)$ : If events $A \ B$ are independent, then $P(A \cap B)=P(A) \times P(B)$.

A
Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$.
B
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A)$.
C
$(A)$ is true but $(R)$ is false.
✓
$(A)$ is false but $(R)$ is true.

✓

Answer

Correct option: D.

$(A)$ is false but $(R)$ is true.

Probability of solving the problem by $A \ B$ is $=1-P\ ($ None of them can solve the problem $)$
$=1-P(\bar{A} \cap \bar{B})=1-P(\bar{A}) \cdot P(\bar{B})$
$=1-[1-P(A)][1-P(B)]$
$=1-\frac{4}{5} \times \frac{3}{5}=\frac{13}{25} .$

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