All three emails are work emails

Question

Work emails arrive as a Poisson process at rate $8$ per hour and personal emails independently at rate $2$ per hour. In a certain stretch, exactly $3$ emails arrived in total.

Given that, what is the probability that all $3$ are work emails?

Accepted Answer

Given the total number of emails, each one is independently work or personal, with probabilities set by the rate shares, so the work-count is binomial. The chance any one email is work is p = \frac{\lambda{	ext{work}}}{\lambda{	ext{work}} + \lambda{	ext{personal}}} = \frac{8}{8 + 2} = 0.8. For all 3 emails to be work, we need this to happen three times independently: P(3) = (0.8)^3 = 0.512. So about a 51\% chance. ("All of one type" is just the extreme case of the binomial, \binom{3}{3}p^3.) Takeaway. "All n arrivals are type i" given the total equals p^n, where p is type i's rate share.

All three emails are work emails

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