A startup that offers two kinds of services, $S_1$ and $S_2$. Both services follow an exponential law with $S_1$ provided, on average, within $1$ minute while $S_2$ within $2$ minutes. $70\%$ of clients prefer $S_1$ and the remaining $30\%$ prefer $S_2$.
- What is the probability that a client will be served in less than $2$ minutes?
- Knowing that the services was provided in less than $2$ minutes, what is the probability that the client asked for $S_1$?
For point 1) I think it is right to use the law of total probability so let A = "the client will be served in less than 2 minutes" then $P(A) = P(A|S_1)P(S_1) + P(A|S_2)P(S_2)$. If this is correct how do I calculate/reason on what is $P(A|S_1)$? For the point 2) requires $P(A)$ (right?) so can I use conditional probability and calculate $P(S_1|A) = \frac{P(A|S_1)P(S_1)}{P(A)}$ ?
Point 1) You are correct so far. To calculate $P(A|S_i)$ note that if $X \sim exp(\lambda)$ then $E[X] = \tfrac{1}{\lambda}$ and you know the average of both $S_1$ and $S_2$.
Point 2) I believe you meant to type the Bayes' rule ($P(X|Y) = \frac{P(Y|X)P(X)}{P(Y)}$) but you made a mistake while typing. If that's the case you are correct again.