The probability that it will rain today is 0.5. The probability that it will rain tomorrow is 0.6. The probability that it will rain either today or tomorrow is 0.7. What is the probability that it will rain today and tomorrow?

The probability that it will rain today is 0.5. The probability that it will rain tomorrow is 0.6. The probability that it will rain either today or tomorrow is 0.7. What is the probability that it will rain today and tomorrow? 

(A) 0.3,    (B) 0.25,    (C) 0.35,    (D) 0.4 

(GATE 1997) 

Answer: (D) 0.4 

Explanation: 

Let $E_1$ be the event of raining today, $E_2$ be the event of raining tomorrow. 

According to the given problem, $P(E_1) = 0.5$, $P(E_2) = 0.6$, and $P(E_1 \cup E_2) = 0.7$. 

Therefore, the required probability is 

$= P(E_1 \cap E_2) $

$= P(E_1) + P(E_2) - P(E_1 \cup E_2)$

$= 0.5 + 0.6 - 0.7 = 0.4$.