# Probability of a patient receiving new treatment

Five percent of patients suffering from a certain disease are selected to undergo a new treatment that is believed to increase the recovery rate from $30$ percent to $50$ percent. A person is randomly selected from these patients after the completion of the treatment and is found to have recovered. What is the probability that the patient received the new treatment?

What I attempted:- Let us define the events

$E$= The event that the selected patient received the new treatment
$\overline{E}$= The patient didn't receive the treatment $A$= The patient has been recovered

We are required to find $P(E|A)$. Here $P(E)=0.05$, $P(\overline{E})=0.95$, $P(A|E)=0.50$, and $P(A|\overline{E})=0.30$

Thus, \begin{aligned} P(E|A)&=\frac{P(E).P(A|E)}{P(E).P(A|E)+P(\overline{E}).P(A|\overline{E})} \\ &=\frac{0.05\times 0.50}{0.05\times 0.50+0.95\times 0.30}\\ &=\frac{5}{62} \end{aligned}

Am I correct ?

Yes, Bayes Theorem is that $P(E|A)$ is $\frac{P(A|E) * P(E)}{P(A)}$ which is exactly what you did.