A box contains 3 different types of flashlights The probabil

A box contains 3 different types of flashlights. The probability that a Type 1 flashlight will give over 100 hours of use is 0.8, the probability that Type 2 gives over 100 hours of use is 0.5, and for Type 3, 0.4. Suppose 50% of the flashlights in the bin are Type 1, 30% are Type 2 and 20% are Type 3.

A) What is the probability that a randomly chosen flashlight will give over 100 hours of use?

B) Given that a flashlight lasted over 100 hours, what is the conditional probability that it was Type 1?

Solution

Let

O = over 100 hrs

Thus,

a)
By Bayes\' Rule,

P(O) = P(1) P(O|1) + P(2) P(O|2) + P(3) P(O|3)

= 0.50*0.8 + 0.30*0.5 + 0.20*0.4

= 0.63 [ANSWER]

*****************

b)

P(1|O) = P(1) P(O|1) / P(O)

= 0.50*0.8/0.63

= 0.634920635 [ANSWER]