Suppose that in one region of the sun’s surface, solar flares erupt according to a Poisson distribution at a mean rate of 24 eruptions per hour.

(a) What is the probability that at least one eruption will occur during the first 15 minutes of an observation period?
(b) What is the probability that there will be more than 10 eruptions during a half-hour observation period?

(a) the mean λ for a period of observation of 15 minutes is λ = 6
(there are, on average, 6 eruptions in 15 minutes).

Let X be the number of eruptions during a given period of 15 minutes.
According to Poisson's law, p(X>0) = 1 - p(X=0) = 1 - exp(-λ) = 1 - exp(-6)

(b) the mean λ for a period of 30 minutes is λ = 12.

p(X>9) = 1 - [ p(X=0) + p(X=1) + ... + p(X=9) ]
= 1 - exp(-12).[ 1 + 12 + 12²/2 + 12^3 / 3! + ... + 12^9 / 9! ]