E-mails per day problem This is an exam problem from probability and statistics that I failed, and still cannot figure out. Thanks in advance.
The number of e-mails that a person receives in a work day (monday to friday) can be modeled with the poisson distribution with an average of 6 e-mails a day. The number of e-mails a person receives during the weekend can also be modeled with the poisson distribution with an average of 3 e-mails a day.
a) What is the probability that person does not receive any e-mails, if we do not know which day it is?
b) A day is chosen at random from the week. If that person didn't receive a an e-mail on that day, what is the probability that day is a work day?
Let $p_1=\exp(-6)$ be the probability of receiving $0$ emails on a week-day and $p_2=\exp(-3)$ be the probability of receiving $0$ emails on a weekend (from the definition of a Poisson random variable). If the day of the week is not known (but all days have the same probability), then the probability of receiving $0$ emails is $\tfrac{5}{7}p_1+\tfrac{2}{7}p_2$.
The probability that today is a workday AND no emails were received is $\tfrac{5}{7}p_1$ so the probability that today is a workday given that no emails were received is $\tfrac{\tfrac{5}{7}p_1}{\tfrac{5}{7}p_1+\tfrac{2}{7}p_2}$.
Let $p_1=\exp(-6)$ be the probability of receiving $0$ emails on a week-day and $p_2=\exp(-3)$ be the probability of receiving $0$ emails on a weekend (from the definition of a Poisson random variable). If the day of the week is not known (but all days have the same probability), then the probability of receiving $0$ emails is $\tfrac{5}{7}p_1+\tfrac{2}{7}p_2$.
The probability that today is a workday AND no emails were received is $\tfrac{5}{7}p_1$ so the probability that today is a workday given that no emails were received is $\tfrac{\tfrac{5}{7}p_1}{\tfrac{5}{7}p_1+\tfrac{2}{7}p_2}$.