The probability of simultaneous occurrence of at least one of two events E and F is \( p \) . If the probability that exactly one of E, F occurs is \( q \) , then \( P(\overline{E})+P(\overline{F}) \) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( q-2p+2 \)
The probability that at least one of E, F occurs is \( P(E\cup F)=p \). “Exactly one” is the union minus the common part:
\( \displaystyle q=P(E\cup F)-P(E\cap F) \)
\( \displaystyle P(E\cap F)=p-q \)
By the addition theorem,
\( \displaystyle P(E)+P(F)=P(E\cup F)+P(E\cap F) \)
\( \displaystyle =p+(p-q)=2p-q \)
Using \( P(\overline{E})=1-P(E) \) and \( P(\overline{F})=1-P(F) \):
\( \displaystyle P(\overline{E})+P(\overline{F})=2-[P(E)+P(F)] \)
\( \displaystyle =2-2p+q \)
\( \displaystyle =q-2p+2 \)