One desktop visit out of four

Question

Visits to a website come from mobile devices as a Poisson process at rate $30$ per hour and from desktops independently at rate $10$ per hour. In a short span, exactly $4$ visits arrived in total.

Given that, what is the probability that exactly $1$ of them is from a desktop?

Accepted Answer

Given the total number of visits, each visit was independently mobile or desktop, with probabilities fixed by the rate shares. So the desktop count is binomial. The chance any one visit is desktop is p = \frac{\lambda{	ext{desktop}}}{\lambda{	ext{mobile}} + \lambda{	ext{desktop}}} = \frac{10}{30 + 10} = 0.25. The number of desktop visits among the 4 is Binomial(4, 0.25). The probability of exactly 1: P(1) = \binom{4}{1}(0.25)^1(0.75)^3 = 4 	imes 0.25 	imes 0.421875 = 0.421875 \approx 0.422. About a 42\% chance. Takeaway. Conditioning on the total turns "how many of each type" into a bino…

One desktop visit out of four

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