GATE CSE 1997 | Question: 1.1

Question

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. $0.4$

Comments

Inclusion-Exclusion Principle (2 Sets)

---

P(A union B) =  P(A) + P(B) – P(A intersection B) , where union is OR and intersection is AND. This will surely help.

Answer 1

Answer: $D$

$P$(it will rain today either today or tomorrow) = $P$(it will rain today) $+ P$(it will rain tomorrow) $- P$(it will rain today and tomorrow)

So, $0.7 = 0.5 + 0.6 - P$(it will rain today and tomorrow)

$\Rightarrow P$ (it will rain today and tomorrow) =$0.4$

Comments

@vivek  ... I agree with you. But then The probability that it will rain either today or tomorrow should be - It rains today but not tomorrow + It does not rain today but rains tomorrow = (0.5) * (0.4) + (0.5) * (0.6) =( 0.5 ) , but in the question it is given 0.7 

@vijaycs

If we consider Events are independent then 

P(A U B) = P(A) + P(B) - P(A)*P(B)

P(it will rain today either today or tomorrow) = P(it will rain today) +P(it will rain tomorrow) −P(it will rain today)*P(it will rain tomorrow) ==> 0.8

how you got 0.5 ..where i am wrong above ??

---

@jatin khachane 1

You are correct, it will be 0.8 only.

@vijaycs has calculated it wrongly for $p(A \oplus B) = p(A \cap \overline{B}) + p(B \cap \overline{A} ) - 2p(A \cap B) $

---

Sometimes we need to forget about real world scenario and just focus on the details that are given in the question. In real world, tomorrow's rain may or may not depend on today's rain, it doesn't matter. In the question, it does depend and that is all we have to think about.

Answer 2

0.7 = 0.5 + 0.6 - p(today and tomorrow) = 0.40

Comments

p(today and tomorrow) should be subtracted instead of being added