If \( P(A)=\displaystyle\frac{2}{5},\ P(\overline{B})=\displaystyle\frac{4}{7} \) and \( P(A\cup B)=\displaystyle\frac{2}{3} \), then which of the following are correct?
(A) \( P(A\cap B)=\displaystyle\frac{17}{105} \)
(B) \( P(A/B)=\displaystyle\frac{17}{42} \)
(C) A and B are independent events
(D) \( P(B/A)=\displaystyle\frac{17}{42} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
Since \( P(\overline{B})=\frac{4}{7} \),
\( \displaystyle P(B)=1-\frac{4}{7}=\frac{3}{7} \)
By the addition theorem, \( P(A\cap B)=P(A)+P(B)-P(A\cup B) \):
\( \displaystyle P(A\cap B)=\frac{2}{5}+\frac{3}{7}-\frac{2}{3} \)
\( \displaystyle =\frac{42+45-70}{105} \)
\( \displaystyle =\frac{17}{105} \)
By the definition of conditional probability:
\( \displaystyle P(A/B)=\frac{17/105}{3/7}=\frac{17}{45} \)
\( \displaystyle P(B/A)=\frac{17/105}{2/5}=\frac{17}{42} \)
For independence, compare with
\( \displaystyle P(A)\,P(B)=\frac{2}{5}\times\frac{3}{7}=\frac{18}{105} \)
So \( P(A\cap B)=\frac{17}{105} \), \( P(A/B)=\frac{17}{45} \), \( P(B/A)=\frac{17}{42} \) and \( P(A)\,P(B)=\frac{18}{105} \).
Checking the statements
(A) \( P(A\cap B)=\displaystyle\frac{17}{105} \) — Correct. \( \frac{2}{5}+\frac{3}{7}-\frac{2}{3}=\frac{17}{105} \).
(B) \( P(A/B)=\displaystyle\frac{17}{42} \) — Incorrect. \( P(A/B) \) is \( \frac{17}{45} \).
(C) A and B are independent events — Incorrect. \( P(A)\,P(B)=\frac{18}{105}\ne\frac{17}{105}=P(A\cap B) \).
(D) \( P(B/A)=\displaystyle\frac{17}{42} \) — Correct. \( \frac{17}{105}\div\frac{2}{5}=\frac{17}{42} \).