$40.$ A box contains ten sealed envelopes numbered $1,...,10.$

The first five contain no money, the next three each con$-$

tains $$5,$ and there is a $$10$ bill in each of the last two. A

sample of size $3$ is selected with replacement $($so we have

a random sample$),$ and you get the largest amount in any

of the envelopes selected. If X$1,$ X$2,$ and X$3$ denote the

amounts in the selected envelopes, the statistic of interest

is M $5$ the maximum of X$1,$ X$2,$ and X$3.$

a$.$ Obtain the probability distribution of this statistic.

b$.$ Describe how you would carry out a simulation

experiment to compare the distributions of M for

various sample sizes. How would you guess the dis$-$

tribution would change as n increases?