# Probability question using bayes'theorem

#### WMDhamnekar

The Department of Health encourages young children to have a flu vaccination each year. The vaccination reduces the likelihood of getting a flu from 40% to 10%. If 45% of the young children visiting the doctor have the vaccination, find the probability that an young child chosen at random had the vaccination, given that he/she get flu.

#### romsek

MHF Helper
This isn't really a Bayes theorem problem.

$P[\text{child chosen at random has flu}] = P[\text{child gets flu|vaccinated}]P[\text{vaccinated}] + P[\text{child gets flu|!vaccinated}]P[\text{!vaccinated}] = \\ (0.1)(0.45) + (0.4)(1-0.45) = 0.265$

math951

#### WMDhamnekar

This isn't really a Bayes theorem problem.

$P[\text{child chosen at random has flu}] = P[\text{child gets flu|vaccinated}]P[\text{vaccinated}] + P[\text{child gets flu|!vaccinated}]P[\text{!vaccinated}] = \\ (0.1)(0.45) + (0.4)(1-0.45) = 0.265$
Hello,
But the author wants the probability that the child is vaccinated given that he/she get flu. You have computed the probability that the child get flu given he/she has been vaccinated.

#### romsek

MHF Helper
$P[\text{vaccinated|got flu}]P[\text{got flu}] = P[\text{got flu|vaccinated}]P[\text{vaccinated}]$
$P[\text{vaccinated|got flu}] = \dfrac{P[\text{got flu|vaccinated}]P[\text{vaccinated}]}{P[\text{got flu}]} = \\ \dfrac{(0.1)(0.45)}{(0.1)(0.45)+(0.4)(1-0.45)} = \dfrac{0.045}{0.265} \approx 0.17$