The Lottery

In a certain state lottery, players try to match all six numbers chosen at random without replacement from a set of 42 numbers.   Prizes are awarded based on how many numbers the player matches.   Players pay $1.00 to play the game.   Suppose that the payoffs for an individual player are as given below, where the grand prize $X is yet to be specified.

Number of
Matches
 Payoff
0, 1, 2
$0
3
$10
4
$100
5
$5000
6
$X

(a) What is the probability distribution for the number of matches?

(b) What is a player's expected number of matches?

(c) What is the minimum value of X for which the game has positive expectation for a player?

(d) Suppose X=$2,000,000, but the grand prize is split evenly among all players who match all six numbers.   If P players try their chances in the lottery including you, find the expectation of the game for you and show that it is a decreasing function of P.