The probability of the simultaneous occurrence of two events A and B is \( p \). If the probability that exactly one of A and B occurs is \( q \), then
A. \( \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2+2q-p \)
B. \( \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2p-q \)
C. \( \mathrm{P}(\overline{\mathrm{A}}\cap\overline{\mathrm{B}})=1-p-q \)
D. \( \mathrm{P}((\mathrm{A}\cap\mathrm{B})|(\mathrm{A}\cup\mathrm{B}))=\displaystyle\frac{p}{p+q} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B, C and D only
Here \( \mathrm{P}(\mathrm{A}\cap\mathrm{B})=p \), and “exactly one occurs” means A only or B only:
\( \displaystyle \left[\mathrm{P(A)}-p\right]+\left[\mathrm{P(B)}-p\right]=q \)
\( \displaystyle \mathrm{P(A)}+\mathrm{P(B)}=q+2p \)
\( \displaystyle \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-(\mathrm{P(A)}+\mathrm{P(B)}) \)
\( \displaystyle =2-2p-q \)
\( \displaystyle \mathrm{P}(\mathrm{A}\cup\mathrm{B})=\mathrm{P(A)}+\mathrm{P(B)}-p \)
\( \displaystyle =p+q \)
By De Morgan's law:
\( \displaystyle \mathrm{P}(\overline{\mathrm{A}}\cap\overline{\mathrm{B}})=1-\mathrm{P}(\mathrm{A}\cup\mathrm{B}) \)
\( \displaystyle =1-p-q \)
Since \( \mathrm{A}\cap\mathrm{B} \) lies inside \( \mathrm{A}\cup\mathrm{B} \):
\( \displaystyle \mathrm{P}((\mathrm{A}\cap\mathrm{B})|(\mathrm{A}\cup\mathrm{B}))=\frac{\mathrm{P}(\mathrm{A}\cap\mathrm{B})}{\mathrm{P}(\mathrm{A}\cup\mathrm{B})} \)
\( \displaystyle =\frac{p}{p+q} \)
So \( \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2p-q \), \( \mathrm{P}(\overline{\mathrm{A}}\cap\overline{\mathrm{B}})=1-p-q \) and \( \mathrm{P}((\mathrm{A}\cap\mathrm{B})|(\mathrm{A}\cup\mathrm{B}))=\frac{p}{p+q} \).
Checking the statements
(A) \( \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2+2q-p \) — Incorrect. The sum is \( 2-2p-q \).
(B) \( \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-2p-q \) — Correct. \( 2-(q+2p)=2-2p-q \).
(C) \( \mathrm{P}(\overline{\mathrm{A}}\cap\overline{\mathrm{B}})=1-p-q \) — Correct. \( 1-\mathrm{P}(\mathrm{A}\cup\mathrm{B})=1-(p+q) \).
(D) \( \mathrm{P}((\mathrm{A}\cap\mathrm{B})|(\mathrm{A}\cup\mathrm{B}))=\frac{p}{p+q} \) — Correct. \( \mathrm{P}(\mathrm{A}\cap\mathrm{B})=p \) divided by \( \mathrm{P}(\mathrm{A}\cup\mathrm{B})=p+q \).