Question

Seventy-seven percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 66% have an emergency locator, whereas 86% 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?

Answer 1

We are given the probabilities here:
P( discovered ) = 0.77
P( have locator | discovered ) = 0.66
P( have locator | not discovered ) = 1 - 0.86 = 0.14

a) Using law of total probability, we get:

P( have locator ) = P( have locator | discovered)P( discovered ) + P( have locator | not discovered ) P( not discovered )

P( have locator ) = 0.66*0.77 + 0.14*(1 - 0.77) = 0.5404

Now using bayes theorem, we have:

P( not discovered | have locator ) = ( have locator | not discovered ) P( not discovered ) / P( have locator )

P( not discovered | have locator )= 0.14*(1 - 0.77) / 0.5404

P( not discovered | have locator )= 0.0596

Therefore 0.0596 is the required probability here.

b) P( no locator ) = 1 - P( locator ) = 1 - 0.5404 = 0.4596

P( discovered | no locator ) = P( no locator | discovered ) P( discovered ) / P( no locator )

P( discovered | no locator ) = (1 - 0.66)*0.77 / 0.4596

P( discovered | no locator ) = 0.5696

Therefore 0.5696 is the required probability here.