Question

There are three types of vaccines A1, A2, A3, available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine A1, 35% of the population was given Vaccine A2 and 40% of the population was given Vaccine A3. The survey also stated that the probabilities that Vaccines A1, A2 and A3 would protect against the infection were 60%, 55% and 50% respectively.

Based on the above information, answer the following questions:

Find the probability that:

1. The person taking vaccine A₂ will get infected. (1)
2. If a person is chosen randomly, he/she will be protected from the infection. (1)
3. 1. The person was given Vaccine A₁, given that the randomly chosen person is infected. (2)
      OR
   2. The person was given Vaccine A₃, given that the randomly chosen person is not infected. (2)

Answer 1

Define the events:

A₁, A₂, A₃: Person was given vaccine A₁, A₂, A₃ respectively.

P(A₁) = 0.25, P(A₂) = 0.35, P(A₃) = 0.40

E: Person is protected

P(E|A₁) = 0.60, P(E|A₂) = 0.55, P(E|A₃) = 0.50

E′: Person is infected (Not protected).

P(E′|A₁) = 0.40, P(E′|A₂) = 0.45, P(E′|A₃) = 0.50

i.

This is the conditional probability P(E′|A₂).

Since 55% are protected, the remaining are infected:

P(E′|A₂) = 1 − 0.55

= 0.45

ii.

We use the Law of Total Probability:

P(E) = P(A₁) P(E|A₁) + P(A₂) P(E|A₂) + P(A₃) P(E|A₃)

= (0.25 × 0.60) + (0.35 × 0.55) + (0.40 × 0.50)

= 0.15 + 0.1925 + 0.20

= 0.5425

iii. (a)

Total probability of being infected:

P(E′) = 1 − P(E)

= 1 − 0.5425

= 0.4575

Now apply Bayes’ Theorem:

P(A1|E′) = `(P(A_1) P(E′|A_1))/(P(E′))`

= `(0.25 xx 0.40)/0.4575`

= `0.10/0.4575`

= 0.2186

iii. b.

Apply Bayes’ Theorem using the total probability of protection from (ii):

P(A₃|E) = `(P(A_3) P(E|A_3))/(P(E))`

= `(0.40 xx 0.50)/0.5425`

= `20/0.5425`

= 0.3687