A plane is landing. If the weather is favourable, the pilot landing the plane can see the runway. In this case the probability of a 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. If the blind landing instruments fail, then the pilot may make a safe landing with 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.

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Solution

$P (s l o p e l a n d i n g) = P$ (safe landing with favorable method ) $+ P (s a f e l a n d i n g w i t h l o w c l o u d t e x t i n g)$
$= (\frac{100 - k}{100}) P_{1} + (\frac{K}{100}) [P . P_{1} + (1 - P) P_{2}]$ ------ $(1)$
$P P_{1} \to$ instrument functions normally and landed safe
$(1-P)P_{2} \rightarrow $ instrument didn't mark and landed safe
$P \frac{(B l i n d l a n d i n g i n s t r u m e n t u s e d)}{s a f e l y l a n d e d} = \frac{P (i n s t r u m e n t u s e d) \times P (\frac{s a f e l y l a n d e d}{i n s t r u m e n t u s e d})}{P (s a f e l y l a n d e d)}$
$= \frac{(\frac{K}{100}) \times [P P_{1} + (1 - P) P_{2}]}{(\frac{K}{100}) P_{1} + (\frac{K}{100}) [P P_{2} + (1 - P) P_{2}]}$

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