(a)[1]
Determine the probability that, in a randomly chosen $2$-day interval, no girls are late.
(b)[3]
Find the probability that, in a randomly selected $5$-day period, the total number of students who arrive late is less than $3$.
(c)[3]
It is given that $\mathrm{P}(G = r)$ and $\mathrm{P}(B = r)$ for $r \geq 3$ are very small and may be ignored. Find the probability that, on a randomly chosen day, more girls are late than boys.
(d)[5]
After a timetable change, the teacher says that, on average, more students are late than before the change. In one randomly selected $5$-day period, a total of $4$ students are late. Test the teacher’s claim at the $5\%$ significance level.