Answer to Question #198412 in Statistics and Probability for bennie

Question

A man is equally likely to choose anyone of the three routes A, B, C from his house to a railway station, and his choice of route is not influenced by weather. If the weather is dry the probabilities of missing a train by routes A, B, C are 1/20, 1/10, 1/5 respectively. He sets out on a dry day and misses the train. a. Find the probability of missing the train . b. Find the probability of choosing route A and missing the train

Answer 1

Let $P(A)$ be the probability of using route A, $P(B)$ be the probability of using route B, $P(C)$ be the probability of using route C, $P(M)$ be the probability of of missing a train.

Given $P(A)=P(B)=P(C)=\dfrac{1}{3}$

$P(M|A)=\dfrac{1}{20}, P(M|B)=\dfrac{1}{10}, P(M|C)=\dfrac{1}{5}$

a.

The Law of Total Probability

$$P(M)=P(A)P(M|A)+P(B)P(M|B)$$

$$+P(C)P(M|C)=\dfrac{1}{3}(\dfrac{1}{20})+\dfrac{1}{3}(\dfrac{1}{10})+\dfrac{1}{5}(\dfrac{1}{3})$$

$$=\dfrac{1+2+4}{60}=\dfrac{7}{60}$$

b. The Multiplication Rule

$$P(C\cap M)=P(C)P(M|C)=$$

$$=\dfrac{1}{3}(\dfrac{1}{20})=\dfrac{1}{60}$$