A school headteacher wants to model the process of children missing school due to illness. There are k children in the school.

The teacher uses the following model, which applies independently to each child and each day:

• If a child is present today, they will be newly absent tomorrow with probability a.

• If a child is absent today, they will be recovered and returned tomorrow with probability r.

ercises 581 There are k = 120 children in the school. Let X be the number of children that are present at school today.

a. Suppose there are 100 children present today. Out of these 100 children, let A be the number that are absent tomorrow, and let P be the number that are present tomorrow. Write down the distributions of A and P.

b. Continue to suppose there are 100 children present today, so there are 20 children absent. Let R be the number of the 20 absentees who return to school tomorrow, and let S be the number that are still absent tomorrow. Write down the distributions of R and S.

c. If 100 children are present today, how many children are present tomorrow? Give your answer as a random variable involving any of the variables A, P, R, and S that you need to use.
You can see where the formula comes from in Exercise 31.13.

d. Let X be the number of children present on a given day.
Although it isn’t obvious, we can show that after the process has been running for a long time, the distribution of X becomes X ∼ Binomial 
k, r r + a 
.
Write down the distribution of X when the recovery probability is r = 0.75, and the new-absence probability is a = 0.05.

e. Using the distribution you found in part (d), write down the R commands for the following probabilities, and calculate them using R.
i. Probability all children are present.
ii. Probability at least 115 children are present.
iii. Probability less than 110 children are present.

f. There are 40 children acting in the school play. In case any of the actors are absent, there are four additional children who can step into their roles. What is the probability that no more than four of the 40 actors will be absent on the day of the play?