A student needs an alarm clock to wake up, which has proven to success

By Bayes Theorem, P(A given B) = P(A/B) = P(A and B) / P(B)


Let the Probability of writing the test = P(W)


To Find: P(Doesn't hear the Alarm given that he doesn't writes the test) = P(A'/W') = P(A' and W') / P(W')


(i) P(W) = P(A) * P(W/A) + P(A') * P(W/A')


(ii) P(A' and W') = 1 - P(A U B) = 1 - [P(A) + P(W) - P(A and W)



Data Given


Probability of hearing the Alarm = P(A) = 0.8

Therefore, Probability of not hearing the Alarm = P(A') = 1 - 0.8 = 0.2


Probability of writing the test given he hears the alarm = P(W/A) = 0.9

Therefore 0.9 = P(W and A) / 0.8

P(W and A) = 0.9 * 0.8 = 0.72


Probability of writing the test given he does not hear the alarm = P(W/A') = 0.5


(i) P(W) = P(A) * P(W/A) + P(A') * P(W/A') = (0.8 * 0.9) + (0.2 * 0.5) = 0.72 + 0.1 = 0.82


Therefore P(W') = 1 - 0.82 = 0.18



(ii) P(A' and W') = 1 - P(A or W) = 1 - [P(A) + P(W) - P(A and W)] = 1 - (0.8 + 0.82 - 0.72) = 0.1



Therefore, P(A'/W') = 0.1 / 0.18 = 10/18 = 5/9



Option C

Arun Kumar