Normal Distribution,standard deviation and probability question.

Question

According a study, the duration of a match in World Cup is approximate normally distributed with the mean 111 minutes and standard deviation 5 minutes (including the break between the halves).

Question: Estimate the probability that more than 30 matches in World Cup 2018 will have the duration between 110 and 118 minutes (assumed there will have 64 matches).

Answer 1

We have $$ X \in N(111, 5) $$ The probability that a single match will last between 110 and 118 minutes is $$ p = P(110 \le X \le 118) = F_X(118) - F_X(110) $$ if $F_X$ is the cummulative dist. function. The probability that at least 30 (out of 64) lasts between these values is $$ 1 - \sum_{i=0}^{30} {64 \choose i}p^i(1-p)^{64-i} $$

Answer 2

Find the probability $p$ that a single match will last between 110 and 118 minutes using the normal distribution. Then use the cumulative binomial distribution to find the probability of more than 30 succeeding.