Question

Consider a particular type of torpedo used on a test range. Some of the torpedoes have been outfitted with a transponder so that after a shot the torpedo can easily be found and reused. Other older torpedoes do not have transponders. Eighty percent of all test range torpedoes are recovered. Of those recovered, 75% have a transponder. Further, of those not recovered, 90 percent do not have a transponder.

Please state the Distribution type and show how to input into excel.

a) Denote recovery as R and failure to recover as R’. Denote having a transponder as T and not have a transponder as T’. What is the probability that a randomly selected torpedo will have a transponder, that is P(T)?

b) What is the probability of not recovering a torpedo given that it has a transponder?

Answer 1

a) The distribution type for the given probabilities is a conditional probability distribution. We can calculate the probability of having a transponder using the law of total probability.

Let's denote:

• R: Recovery of torpedo

• R': Failure to recover torpedo

• T: Torpedo has a transponder

• T': Torpedo does not have a transponder

We are given the following probabilities:

• P(R) = 0.8 (Probability of recovering a torpedo)

• P(R') = 1 - P(R) = 0.2 (Probability of not recovering a torpedo)

• P(T|R) = 0.75 (Probability of having a transponder given that it is recovered)

• P(T'|R') = 0.9 (Probability of not having a transponder given that it is not recovered)

To calculate the probability of a randomly selected torpedo having a transponder, P(T), we can use the law of total probability:

P(T) = P(T|R) * P(R) + P(T|R') * P(R') = 0.75 * 0.8 + 0 * 0.2 = 0.6

Therefore, the probability that a randomly selected torpedo will have a transponder is 0.6.

b) We need to find the probability of not recovering a torpedo given that it has a transponder, P(R'|T).

Using Bayes' theorem, we have:

P(R'|T) = (P(T|R') * P(R')) / P(T) = (0 * 0.2) / 0.6 = 0

Therefore, the probability of not recovering a torpedo given that it has a transponder is 0.

Answer 2