Question

A certain company sends 35% of its overnight mail parcels via express delivery service 1 and the rest by express delivery service 2. Express delivery service 1 has a record of 3.0% of packages being delivered late and express delivery service 2 has a record of 2.0% of packages being delivered late. A package was delivered late. What is the probability that a late package was delivered by express delivery service 2

Answer 1

55.32% probability that a late package was delivered by express delivery service 2

Explanation:

Bayes Theorem:

Two events, A and B.

`P(B|A) = (P(B)*P(A|B))/(P(A))`

In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.

In this question:

Event A: Late delivery.

Event B: Service 2 was used.

A certain company sends 35% of its overnight mail parcels via express delivery service 1 and the rest by express delivery service 2.

100 - 35 = 65%.

So
`P(B) = 0.65`

Service 2 has a record of 2.0% of packages being delivered late.

This means that
`P(A|B) = 0.02`

Probability of a late delivery.

35% from service 1. Of those, 3% are late.

65% from service 2. Of those, 2% are late.

So

`P(A) = 0.35*0.03 + 0.65*0.02 = 0.0235`

What is the probability that a late package was delivered by express delivery service 2

`P(B|A) = (0.65*0.02)/(0.0235) = 0.5532`

55.32% probability that a late package was delivered by express delivery service 2