Probability Question - CBSE

Question

A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\dfrac{3}{10}, \dfrac{1}{5}, \dfrac{1}{10}$ and $\dfrac{2}{5}$. The probabilities that he will be late are $\dfrac{1}{4}, \dfrac{1}{3}$ and $\dfrac{1}{12}$ if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. What is the probability that he came by train?

Accepted Answer

Priors: $P(E_1)=3/10, P(E_2)=1/5, P(E_3)=1/10, P(E_4)=2/5$. [1.0 Mark]
Likelihoods: $P(L|E_1)=1/4, P(L|E_2)=1/3, P(L|E_3)=1/12, P(L|E_4)=0$. [1.5 Marks]
Denominator $P(L) = 3/40 + 1/15 + 1/120 = 18/120 = 3/20$. [1.5 Marks]
$P(E_1|L) = \dfrac{3/40}{3/20} = \dfrac{1}{2}$. [1.0 Mark]

---
🎯 Official CBSE Marking Scheme:
Evaluating transport probabilities: 1.0 Mark
Evaluating conditional lateness probabilities: 1.5 Marks
Evaluating total probability of lateness $P(L) = 3/20$: 1.5 Marks
Evaluating final probability $= 1/2$: 1.0 Mark

Question:

Step-by-Step Solution

Master Probability with Mathbee