Question

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A, B) = $\frac{5}{9}$, then p = _________.

Answer 1

Since A and B are independent
⇒ $\overline{A}$ and B are independent and A and $\overline{B}$ are independent
Given P(A) = p
i.e P$\left(\overline{A}\right)$ = 1 – p
P(B) = 2p
i.e P$\left(\overline{B}\right)$ = 1 – 2p and P(exactly one of A, B) = $P\left(\overline{A}\cap B\right)+P\left(A\cap \overline{B}\right)$
$\frac{5}{9}=P\left(\overline{A}\right)P\left(B\right)+P\left(A\right)P\left(\overline{B}\right)\left[\because \left(\overline{A}·B\right)\mathrm{and}\left(A·\overline{B}\right)\mathrm{pairs}\mathrm{are}\mathrm{also}\mathrm{independent}\right]$
i.e $\frac{5}{9}$ = (1 – P) (2p) + P(1– 2p)
= 2p – 2p² + p – 2p²
i.e $\frac{5}{9}$ = 3p – 4p²
i.e 5 = 27p – 36p²
i.e 36p² – 27p + 5 = 0
i.e 36p² – 27p + 5 = 0
i.e D = (27)² – 4(36)(5)
i.e D = 729 – 720
i.e D = 9
$$\begin{gathered} \therefore p=\frac{+27\pm \sqrt{9}}{2\times 36} \\ =\frac{+27\pm 3}{2\times 36} \\ =\frac{+30}{2\times 36},\frac{24}{2\times 36} \\ \mathrm{i}.\mathrm{e}p=\frac{5}{12},\frac{1}{3} \end{gathered}$$