5 goals are scored in a 60-min hockey game (ignore breaks), and follow a Poisson process. What is the probability that at least 3 goals are scored in the last quarter of the game?
I know when we have the total number of goals in an interval, the "arrival" time of each one is uniformly distributed, $U(0,60)$. However, I cannot solve this analytically.
I know based on simulation that the answer is very close to 0.1038.
Any guidance will be highly appreciated.
You are correct! Each goal under these assumptions is uniformly distributed on (0,60). Therefore the probability that any one of them is in the last quarter is 0.25=1/4, and hence the number of goals $X_5$ in the fourth quarter is distributed as a Binomial random variable with parameters $n=5$ and $p=0.25$ (why?). Hence $P(X\ge 3) = 1-P(X\le 2) = 1 - {5 \choose 2} 0.25^2 0.75^3-{5 \choose 1} 0.25^1 0.75^4- {5 \choose 0} 0.25^0 0.75^5 = 0.1035156$.