Seventy-eight percent of the light aircraft that disappear while in flight in a

Question

Seventy-eight percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 62% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared. (Round your answers to three decimal places.)

(a) If it has an emergency locator, what is the probability that it will not be discovered?


(b) If it does not have an emergency locator, what is the probability that it will be discovered?

Explanation / Answer

Let A be the event of aircraft being discovered and B be the event of having an emergency locator.

P(A) = 0.78 P(B|A) = 0.62 P(Bc|Ac) = 0.9

P(A B) = P(B|A) P(A) = 0.78 * 0.62 = 0.4836

P(Ac) = 1 - 0.78 = 0.22

P(A U B)c = P(Ac Bc) = P(Bc|Ac) P(Ac) = 0.9 * 0.22 = 0.198

=> P(A U B) = 1 - 0.198 = 0.802

P(B) = P(A U B) + P(A B) - P(A)

= 0.802 + 0.4836 - 0.78 = 0.5056

(a) P(Ac|B) = 1 - P(A|B) = P(A B) / P(B)

= 1 - 0.4836 / 0.5056

= 1 - 0.9565

= 0.0435.

(b) P(Bc) = 1 - P(B)

= 1 - 0.5056 = 0.4944

P(A Bc) = P(A) - P(A B)

= 0.78 - 0.4836

= 0.2964

P(A|Bc) = P(A Bc) / P(Bc)

= 0.2964 / 0.4944

= 0.5995.

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