Roads A, B, and C are the only way to escape from a certain provincial prison. P

Question

Roads A, B, and C are the only way to escape from a certain provincial prison. Prison records show that, of the prisoners who tried to escape, 3 % used road A, 25 % used road B, the remainder used road C. The records also indicate that 53 % of those who tried to escape using road A were captured. 10 % of those using road B were captured, and 56 % of those using road C were captured. Use four decimals in your answers.

(a) What is the probability that a prisoner escaping from this provincial prison is not captured?  

(b) What is the probability that a captured prisoner used road A in their escape attempt?

(c) What is the probability that a prisoner who didn't get captured has used road C?

Explanation / Answer

Let

A, B , C = the roads they used

X = captured

a)

P(X) = P(A) P(X|A) + P(B) P(X|B) + P(C) P(X|C)

P(X) = 0.03*(0.53) + 0.25*(0.10) + (1-0.03-0.25)*(0.56)

P(X) = 0.4441

Hence,

P(not X) = 1 - 0.4441 = 0.5559 [ANSWER]

******************

b)

P(A|X) = P(A) P(X|A) / P(X)

= 0.03*0.53/0.4441 = 0.035802747 [ANSWER]

******************

c)

P(C|not X) = P(C) P(not X|C) / P(not X)

= (1-0.03-0.25)*(1-0.56)/(0.5559)

= 0.56988667 [ANSWER]

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