Based on above information given below, answer
A doctor is to visit a patient. From the past experiences, it is known that the probabilities of his coming by cab, metro, bike or by other means of transport are respectively 0.3, 0.2, 0.1 and 0.4 respectively. The probabilities that he will be late are 0.25, 0.3, 0.35 and 0.1 if he comes by cab, metro, bike and by other means of transport respectively. Using this information about the doctor, answer the following questions:
When the doctor arrives late, what is the probability that he comes by bike?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{6}\)
Let \(E_1, E_2, E_3, E_4\) be the events that the doctor comes by cab, metro, bike and other means, and L the event that he is late. By the theorem of total probability:
\(P(L) = 0.3(0.25) + 0.2(0.3) + 0.1(0.35) + 0.4(0.1)\)
\( = 0.075 + 0.06 + 0.035 + 0.04 = 0.21\)
By Bayes’ theorem:
\( \displaystyle P(E_3 \mid L) = \frac{P(E_3)\,P(L \mid E_3)}{P(L)} = \frac{0.035}{0.21} = \frac{1}{6} \)