You lost your keys in one of classrooms 1, 2, 3. or 4. You believe that they are

Question

You lost your keys in one of classrooms 1, 2, 3. or 4. You believe that they are in classrooms 1, 2, or 3 with probability 0. 3 for each, or in classroom 4 with probability 0. 1, You have just enough time to search three classrooms, but your friend Shannon has offered to help you look Given your past experience, you know that Shannon has a probability 0. 6 and you have a probability 0 8 of finding the keys when searching in the room that has the keys You are twice as fast as Shannon. (a) What is the probability of finding the keys if you looked in classrooms 1 and 2, while Shannon searched 3, and classroom 4 remained unchecked? (b) What is the conditional probability that the keys are in room 3, given that the keys were not found after the search described in (a)?

Explanation / Answer

Answer to part a)

There are two scenarios : either I get the key or Shannon gets it

So the probability = P(Me) + P(shannon)

P(Me) = 0.8* (0.3+0.3)= 0.8*0.6 = 0.54

P(Shannon) = 0.6*0.3 = 0.18

Thus total Probability = 0.54 +0.18 = 0.72

.

Answer to part b)

P( keys in room # 3 given not found as described in part a) = P(keys in room# 3) / P(keys not found as described)

P( not found) = 1 - 0.72 = 0.18

P(key is in room# 3 , but shannon didnot find it) = 0.3 * (1-0.6)

[P(keys in the room# 3) = 0.3

P(shanon fails to find the key) = 1-0.6 = 0.4]

P(keys in room# 3, but Sannon didnot find it ) = 0.12

.

Thus conditional probability = 0.12/0.18

Condiitonal Probability = 0.6667

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