Problem 2: Probability and Bayes Rule (30 pts) Seventy percent of the light airc

Question

Problem 2: Probability and Bayes Rule (30 pts) Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose an aircraft has disappeared. Consider the two events D: The disappeared aircraft is discovered, and L the disappeared aircraft has a locator. a. (10 pts) That is the probability that a disappeared aircraft will have a locator? b. (10 pts) If it has an emergency locator, what is the probability that it will not be discov- ered? c. (10 pts) Are the events being discovered and having a locator independent?

Explanation / Answer

Solution:-

Given, P(D) = 0.70

P(L I D) = 0.6 (Of the aircrafts that are discovered, 60% have an emergency locator)

P(NL I ND) = 0.9 (90% of the aircrafts not discovered do not have such a locator)

a). Pr(Disappeared have locator) =

  P(L) = P(L I D)P(D) + P(L I ND)P(ND)

P(L I ND) = 1 - P(NL I ND) = 1 - 0.90 = 0.10

Therefore,

P(L) = (0.6)(0.7) + (0.1)(0.3) = 0.45

b). P(ND I L) = P(L I ND) * P(ND) / P(L)

P(ND) = 1 - P(D) = 1 - 0.70 = 0.30

P(L) = P(L I D)P(D) + P(L I ND)P(ND)

P(L I ND) = 1 - P(NL I ND) = 1 - 0.90 = 0.10

Therefore,

P(L) = (0.6)(0.7) + (0.1)(0.3) = 0.45

Thus, P(NL) = 0.55

P(ND I L) = (0.1)(0.3) / (0.45) = 0.067

c). No the events being discovered and having a locator are not independent,

As an independent event is an event that has no connection to another event's chances of happening (or not happening). In other words, the event has no effect on the probability of another event occurring. When two events are independent, one event does not influence the probability of another event.

And here as we have seen, that an aircraft with locator has higher probability of being discovered.

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