In Kentucky$$s Pick $4$ game, you pay $$1$ to select a sequence of four

digits, such as $2283.$ If you buy only one ticket and win, your prize is $$5000$ and your net gain

is $$4999.$

a$.$ If you buy one ticket, what is the probability of winning?

b$.$ Construct a table describing the probability distribution corresponding to the purchase of

one Pick $4$ ticket.

c$.$ If you play this game once every day, find the mean number of wins in years with exactly

$365$ days.

d$.$ If you play this game once every day, find the probability of winning exactly once in $365$ days.

e$.$ Find the expected value for the purchase of one ticket.