ISRO 2012: [Mech] Probability

Question

Let E and F be any two events with P(E U F)= 0.8, P(E) = 0.4 and P (E/ F) = 0.3. Then P (F) is

(a) 3/7
(b) 4/7
(c) 3/5
(d) 2/5

Answer 1

Let P(F) be x.

P(E/F) = P(E $\bigcap$ F) / P(F) which gives P(E $\bigcap$ F) = 0.3*x

Now, 0.8 = P(E) + P(F) - 0.3*x

0.4 = x(1-0.3) which gives P(F) = 0.4/0.7 = 4/7 (option B)

Answer 2

$$\begin{align*} &\Rightarrow p(E/F) = \frac{p(E \cap F)}{p(F)} \\ &\Rightarrow p(E/F) = \frac{p(E) + p(F) - p(E \cup F)}{p(F)} \\ &\Rightarrow p(E/F) = 1+ \frac{p(E) - p(E \cup F)}{p(F)} \\ &\Rightarrow 0.3 - 1 = \frac{p(E) - p(E \cup F)}{p(F)} \\ &\Rightarrow p(F) = \frac{0.4 - 0.8}{-0.7} \\ &\Rightarrow p(F) = \frac{4}{7} \\ \end{align*}$$