Q:

Bayes theorem and conditional probability

I have a problem like this: 

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, 60% have an emergency locator, whereas
  85% 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?

I know D = discovered = .78, and so D-complement = not discovered = .22
Thanks

A:

If we define:
$A$ - aircraft that are discovered.
$B$ - aircraft that have an emergency locator.
We know from the question that: $P(A) = 0.78$, $P(B\mid A) = 0.6$ and $P(B_c\mid A_c) = 0.85$. so using Bayes theorem we can deduce that:
$$ \frac{P(B\cap A)}{P(A)} = 0.6 \Longrightarrow P(B\cap A) = 0.468$$
$$ \frac{P(B_c\cap A_c)}{P(A_c)} = 0.85 => P(B_c\cap A_c) = 0.187$$ 
I hope you can take it from here.  

