Chris Y. answered • 05/14/19

State-certified math teacher with 5+ years experience

The probability of a single poisson event is e^{-λ}λ^{k}/k!, where λ is the mean of the distribution (3 in this case).

The policy pays nothing on 0, 1, or 2 occurrences and pays 50,000 for each occurrence afterward. So we want the probability of 0, 1, or 2 occurrences and the probability of everything else or

P(N < 3) and

P(N > 2) = 1 - P(N < 3)

And our expected value is each probability multiplied by the corresponding payout:

E(X) = 0 * P(N < 3) + 50,000 * (1 - P(N < 3))

So we really only need P(N < 3). Use the formula for the poisson probability above.

P(N < 3) = P(N=0) + P(N=1) + P(N=2)

P(N = 0) = e^{-3}3^{0}/0! = e^{-3}

P(N = 1) = e^{-3}3^{1}/1! = 3e^{-3}

P(N = 2) = e^{-3}3^{2}/2! = 9/2 * e^{-3}

So P(N < 3) = e^{-3} + 3e^{-3} + 9/2 * e^{-3} = 17/2 * e^{-3}

Then plug into our expected value formula above:

E(X) = 0 * (17/2 * e^{-3}) + 50,000 * (1 - 17/2 * e^{-3}) ≈ **$28,840.50**