Question

Three airlines serve a Srinagar. Airline ‘Amira’ has 50% of all the scheduled flights, airline ‘Biyas’ has 30%, and airline ‘chinar’ has the remaining 20%. Their on-time rates are 80%, 65%, and 40%, respectively.

Part 1) Draw the Probability tree diagram. (Note: You may use any software for this, like MS paint, MS office, etc.)

Part 2) A plane has just left on time. What is the probability that it was airline ‘Amira’?

Answer 1

To calculate the probability that it was airline A, we consider the following information as interpreted from the probability tree diagram.

Event A represents that scheduled flights for Airline A.

Event B represents that scheduled flights for Airline B.

The event C represents that scheduled flights for Airline C.

The event E represents that flight left in on time.

The prior probabilities are:

P(A)=0.50

P(B)=0.30

P(C)=0.20

The posterior probabilities are:

P(E|A)=0.80

P(E|B)=0.65

P(E|C)=0.40

We have to find the probability that it was airline A if the plane has just left on time.

That is, we have to find P(A|E)

Substituting the prior and likelihood (posterior) probabilities into the Bayes’s Law formula, then it yields

$P(A|E) = \frac{P(E|A) \times P(A)}{P(E|A) \times P(A) + P(E|B) \times P(B) + P(E|C) \times P(C)} \\ = \frac{0.80 \times 0.50}{0.80 \times 0.50 + 0.65 \times 0.30 + 0.40 \times 0.20} \\ = \frac{0.4}{0.675} \\ = 0.59259$

Therefore, the probability that it was airline A if the plane has just left on time is 0.593.