4. A Prisoner escapes from jail. There are three roads leading away form the jail. If the prisoner selects road A to make good here escape, the probability that she succeeeds is 1/4. If she selects road B, the probability that she succeeds is 1/5. If she selects road C, the probability she succeeds is 1/6. Furthermore, the prpbability that she selects of these roads is the same, Is is 1/3rd. If the prisoner succeeds on her escape, what is the probability that she made good her escape by using road B?
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To find the probability that the prisoner made a successful escape using road B, we need to use Bayes' Theorem.
Bayes' Theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
Where:
P(A|B) is the probability of event A given that event B has occurred,
P(B|A) is the probability of event B given that event A has occurred,
P(A) is the probability of event A,
P(B) is the probability of event B.
In this case:
Event A is the prisoner making a successful escape using road B,
Event B is the prisoner making a successful escape overall.
We are given that the probability of the prisoner successfully escaping using road B is 1/5, and the probability of the prisoner successfully escaping overall is 1/3.
To calculate the probability that the prisoner made a successful escape using road B, we need to find P(B|A). This is the probability of the prisoner making a successful escape overall given that she used road B.
Using Bayes' Theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
Substituting the given values:
P(A|B) = (1/5 * 1/3) / (1/3)
Calculating the expression, we get:
P(A|B) = 1/5
Therefore, the probability that the prisoner made good her escape by using road B is 1/5.