A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate

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Correct answer: B. (12)/(37)

Let E_1 , E_2 , and E_3 be the events that the candidate travels by bus, scooter, and car, respectively. The probabilities of choosing these modes of transport are: P(E_1) = (2)/(5) P(E_2) = (1)/(5) P(E_3) = (2)/(5) Let A be the event that the candidate reaches the examination centre late. The conditional probabilities of reaching late are: P(A|E_1) = (1)/(5) P(A|E_2) = (1)/(3) P(A|E_3) = (1)/(4) We need to find the probability that the candidate travelled by bus given that they reached late, which is P(E_1|A) . Using Bayes' theorem: P(E_1|A) = (P(E_1)P(A|E_1))/(P(E_1)P(A|E_1) + P(E_2)P(A|E_2) + P(E_3)P(A|E_3)) Substituting the given values: P(E_1|A) = ((2)/(5) × (1)/(5))/(((2)/(5) × (1)/(5)) + ((1)/(5) × (1)/(3)) + ((2)/(5) × (1)/(4))) P(E_1|A) = ((2)/(25))/((2)/(25) + (1)/(15) + (1)/(10)) Taking the LCM of the denominators 25 , 15 , and 10 , which is 150 : P(E_1|A) = ((12)/(150))/((12)/(150) + (10)/(150) + (15)/(150)) P(E_1|A) = (12)/(12 + 10 + 15) = (12)/(37) Answer: (12)/(37)

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