# Mr. X goes to office by Auto, Car, and train. The probabilities him travelling by these modes are 27,37,27 respectively. The chances of him being late to the office are 12,14,14 respectively - Mathematics and Statistics

Sum

Mr. X goes to office by Auto, Car, and train. The probabilities him travelling by these modes are 2/7, 3/7, 2/7 respectively. The chances of him being late to the office are 1/2, 1/4, 1/4 respectively by Auto, Car, and train. On one particular day, he was late to the office. Find the probability that he travelled by car.

#### Solution

Let A, C, and T be the events that Mr. X goes to office by Auto, Car, and Train respectively. Let L be event that he is late.

Given that P(A) = 2/7, P(C) = 3/7, P(T) = 2/7

"P"("L"/"A") = 1/2, "P"("L"/"C") = 1/4, "P"("L"/"T") = 1/4

P(L) = P(A ∩ L) + P(C ∩ L) + P(T ∩ L)

="P"("A")*"P"("L"/"A") + "P"("C")*"P"("L"/"C") + "P"("T")*"P"("L"/"T")

= 2/7 xx 1/2 + 3/7 xx 1/4 + 2/7 xx 1/4

= 4/28 + 3/28 + 2/28 = 9/28

"P"("C"/"L") = ("P"("L" ∩ "C"))/("P"("L"))

= ("P"("C")"P"("L"/"C"))/("P"("L"))

= (3/7 xx 1/4)/(9/28

= 1/3

Concept: Baye'S Theorem
Is there an error in this question or solution?