Answer to Question #235408 in Statistics and Probability for Olu

Question

The prisoner picks an exit a random .if he did make a successful escape, what is the probability that the escape was made via exit A

i.0.333
ii.0.571
iii.0.133
iv.0.429
v.0.364

Answer 1

Total problem: A prisoner is considering his chances of escape. From his cell, there are 3 possible exists he could choose (exit A, exit B, exit C). From these exits, the chances of successfully escaping are 0.4, 0.3, and 0.4 respectively. The prisoner picks an exit at random. If he did make a successful escape, what is the probability that the escape was made via exit A (final calculation to 3 decimal places)?

P(exit A) = P(exit B) = P(exit C) =$\frac{1}{3}$

P(Successful escape | exit A) = 0.4

P(Successful escape | exit B) = 0.3

P(Successful escape | exit C) = 0.4

There are three possible exits.

Hence, the probability of choosing any one exit will be $\frac{1}{3}.$

Now, the probability will make a successful escape if he either chooses exit A and escape successfully or chooses exit B and escape successfully or chooses exit A and escape successfully.

P(Escaped successfully) $= \frac{1}{3} \times (0.4 + 0.3 + 0.4)$

P(escaped successfully) $= \frac{1.1}{3}$

P(Escaped successfully) = 0.367

Hence, the probability that the prisoner will make a successful escape is 0.367.

We have to find P(exit A | successful escape).

P(exitA|successful escape) $= \frac{exit \;A\; and \;successful \;escape}{P(successful \;escape)}$

P(exitA|successful escape) $= \frac{P(sucessful \;escape | exit \;A) \times P(exit \;A)}{P(successful \;escape)}$

$= \frac{0.4 \times (1/3)}{0.367} = 0.364$