GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 21

Question

Suppose we have events $A, B$ in a sample space. And we know that $\mathrm{P}(\mathrm{A})=0.3, \mathrm{P}\left(\mathrm{B} \mid \mathrm{A}^c\right)=0.25, \mathrm{P}(\mathrm{B} \mid \mathrm{A})=0.45$. 

What is $\mathrm{P}\left(\mathrm{A}^c \mid \mathrm{B}\right) ?$

• A. 0.75
• B. 0.55
• C. 0.2
• D. 0.56

Answer 1

I hope this finds helpful...

Answer 2

This question can be solved simply using the concepts of Baye's Theorem in conditional probability.

The following data is given: -

P(A) = 0.3, P(B | A^c) = 0.25 and P(B | A) = 0.45

From the given data, we can find out: -

P(A^c) = 1- P(A) = 1- 0.3 = 0.7

Also, P(B) = [P(A) x P(B | A)] + [P(A^c) x P(B | A^c)] = [0.3 x 0.45] + [0.7 x 0.25] = 0.135 + 0.175 = 0.31

Now, according to Bayes' Theorem: -

P(A^c | B) = [P(B | A^c) x P(A^c)] / P(B) = [0.7 x 0.25] / 0.31 = 0.175 / 0.31 = 0.564 ≈ 0.56

Therefore, the correct answer is Option (D).