Question

Seventy percent of 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 an emergency locator. Suppose that a light aircraft has disappeared.
If it has an emergency locator, what is the probability that it will not be discovered?

Answer 1

Answer:

If it has an emergency locator, there is a 6.67% probability that it will not be discovered.

Step-by-step explanation:

This can be formulated as the following problem:

What is the probability of B happening, knowing that A has happened.

It can be calculated by the following formula

$P = \frac{P(B).P(A/B)}{P(A)}$

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

For this problem, we have that:

What is the probability of the aircraft not being discovered, given that it has an emergency locator.

P(B) is the probability that the aircraft is not discovered. Seventy percent of light aircraft that disappear while in flight in a certain country are subsequently discovered. So 30% are not discovered. This means that $P(B) = 0.3$.

P(A/B) is the probability that the aircraft has an emergency locator, given that it was not discovered. 90% of the aircraft not discovered do not have an emergency locator.  So, there is a 10% probability that an aircraft that was not discovered has an emergency locator. This means that $P(A/B) = 0.10$

P(A) is the probability that an aircraft has an emergency locator. 70% of the aircrafts are located, of which 60% have an emergency locator. 30% of the aircrafts are not located, of which 10% have an emergency locator. So

$P(A) = 0.7(0.6) + 0.3(0.1) = 0.45$

Finally

$P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.3*0.1}{0.45} = 0.0667$

If it has an emergency locator, there is a 6.67% probability that it will not be discovered.