For two events A and  B, P(A) = P(A|B) = 0.25 and P(BIA) = 0.5. Which of the following are correct?

I. A and B are independent.

II. P(A^c ∪ B^c) = 0.875

III. P(A^c ∩ B^c) = 0.375

Select the answer using the code given below.

Explanation:

Given:

\(P(A) = P(\frac{A}{B}) = 0.25\)

and \(P(\frac{B}{A}) = 0.5\)

I. \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(A)}\)

⇒ P(A∩B) = P(A) P(B|A)

⇒ P(A∩B) = 0.25 × 0.5 = 0.125

Now

⇒ \(P(\frac{B}{A}) = \frac{P(A∩ B)}{P(B)}\)

⇒ \(P(B)= \frac{P(A∩ B)}{P(\frac{A}{B})}\)

⇒ \(P(B) = \frac{0.125}{0.25} = 0.5\)

Now, P(A).P(B) = 0.25 × 0.5 = 0.125 = P(A∩B)

Thus  A and B are independent

II. \(P(\overline A\cup \overline B ) = 1 – P(A ∩ B)\)

= 1 – 0.125 = 0.875

III. \(P(\overline A∩ \overline B ) = 1 – P(A \cup B)\)

= 1 – [P(A) + P(B) – P(A ∩ B)

=  1 – [0.25 + 0.5 – 0.125]

= = 1 – 0.625 = 0.375

So all statements I, II, and III are correct.

∴ Option (d) is correct.