Question

An influenza vaccine is produced by two different companies. It is known that a vaccine produced by company 1 is effective with probability 0.89, while a vaccine produced by company 2 is effective with probability 0.93. We also know that company 1 supplies 40% of the vaccines, while company 2 supplies 60% of the vaccines ordered by the government.

(i) What is the probability that a vaccine is effective, given that it was produced by company 2?

(ii) What is the probability that a randomly chosen vaccine from the government’s order is not effective?

(iii) What is the probability that given a vaccine is not effective that it was produced by company 1?

Answer 1

Define the following events,

Let $C_1$ and $C_2$ be the events that the influenza vaccine is manufactured by company 1 and company 2 respectively. Also, let $E$ be the event that the influenza vaccine produced is effective and $E'$ is the event that the influenza vaccine produced is not effective.

The following probabilities are given,

$p(E|C_1)=0.89,\space p(E|C_2)=0.93,\space p(C_1)=0.40,\space p(C_2)=0.60$

$i)$

The probability that a vaccine is effective, given that it was produced by company 2 is given as,

$p(E|C_2)=0.93$.

$ii)$

We determine the probability that a randomly selected vaccine is effective. To do so, we shall apply the law of total probability as follows,

$p(E)=p(E|C_1)*p(C_1)+p(E|C_2)*p(C_2)=0.89*0.40+0.93*0.60=0.356+0.558=0.914$

The probability that a randomly selected vaccine is not effective is given as,

$p(E')=1-p(E)=1-0.914=0.086$

Therefore, the probability that a randomly selected vaccine is not effective is 0.086.

$iii)$

Here, we determine the conditional probability, $p(C_1|E')$ defined as,

$p(C_1|E')={p(C_1\cap E')\over p(E')}$

$P(C_1\cap E')=0.4\times0.11=0.044\space and \space p(E')=0.086$

Thus, $p(C_1|E')={0.044\over 0.086}=0.5116(4dp)$

Therefore, the probability that a randomly selected vaccine is produced by company 1 given it is not effective is 0.5116.