Question

A plane is landing. If the weather is favorable, the pilot landing the plane can see the runway. In this case, the probability of safe landing is ${P_1}$. If there is a low cloud ceiling, the pilot has to make a blind landing by instruments. The reliability (the probability of failure-free functioning) of the instruments needed for a blind landing is $P$. If the blind landing instruments function normally, the plane makes a safe landing with the same probability ${P_1}$ as in the case of a visual landing instruments fail, then the pilot may make a safe landing with the probability ${P_2} < {P_1}$. Compute the probability of a safe landing if it is known that in $K$ percent of the cases there is a low cloud ceiling. Also, find the probability that the pilot used the blind landing instrument if the plane landed safely.

Answer 1

At first, we will find the probability of safe landing with favorable conditions and with the help of the found probability we will find the probability of safe landing under a low cloud ceiling, then we will find the conditional probability of the safe landing using the instruments with the above-found probabilities.

Complete step by step solution:
It is given that, if the weather is favorable, the pilot landing the plane can see the runway. In this case, the probability of safe landing is ${P_1}$.
Again, the reliability (the probability of failure-free functioning) of the instruments needed for a blind landing is $P$ and if the blind landing instruments function normally, the plane makes a safe landing with the same probability ${P_1}$ as in the case of a visual landing instruments fail, then the pilot may make a safe landing with the probability ${P_2} < {P_1}$.
The probability of a safe landing if it is known that in $K$ percent of the cases there is a low cloud ceiling.
So, we have, the instrument functions normally and landed safely for $P{P_1}$ times.
And, the instrument did not mark and landed safely for $(1 - P){P_2}$ times.
From these we have,
Prob (safe landing with favourable condition) $= \dfrac{{100 - k}}{{100}}{P_1}$
Prob (safe landing with low cloud ceiling) $= \dfrac{K}{{100}}[P.{P_1} + (1 - P){P_2}]$
Prob (safe landing) $= \dfrac{K}{{100}}{P_1} + \dfrac{K}{{100}}[P{P_2} + (1 - P){P_2}]$
Therefore,
Prob (safe landing) = prob (safe landing with favourable condition) + prob (safe landing with low cloud ceiling)
$= \dfrac{{100 - k}}{{100}}{P_1} + \dfrac{K}{{100}}[P.{P_1} + (1 - P){P_2}]$
Prob (the pilot used the blind landing instrument) = prob (safe landing with low cloud ceiling) $\div$prob (safe landing)
$= \dfrac{{\dfrac{K}{{100}} \times [P{P_1} + (1 - P){P_2}]}}{{\dfrac{K}{{100}}{P_1} + \dfrac{K}{{100}}[P{P_2} + (1 - P){P_2}]}}$

Note:
It is known that in $K$ percent of the cases there is a low cloud ceiling, then the chances for a safe landing are $\dfrac{{100 - K}}{{100}}$percent. Here while finding the probabilities we should initially find the possibility of both the events to happen.