The probability that a plant A survives is $\frac{3}{4}$ and the probability that another plant B survives is $\frac{1}{3}$. The probability that only one of them survives is :
Answer & explanation
Correct answer: option 4
Let A → plant A survives
B → plant B survives
$P(A)=3 / 4$
$P(B)=1 / 3$
$P(A \cap B)=P(A) P(B)$
$=\frac{3}{4} \times \frac{1}{3}$
$P(A \cap B)=1 / 4$
$A \cap B \rightarrow$ Plant $A$ and $B$ both survive
$A \cup B \rightarrow$ Plant $A$ or $B$ survives
$P(\text { required })=P(A \cup B)-P(A \cap B)$
as $P(A \cup B) = P(A)+P(B) - P(A \cap B)$
$=P(A)+P(B) \leq P(A \cap B)-P(A \cap B)$
$=P(A)+P(B)-2 P(A \cap B)$
$=\frac{3}{4}+\frac{1}{3}-2 \times \frac{1}{4}$
$=\frac{9+4-6}{12}=\frac{7}{12}$