In the Michigan lottery game, LOTTO 47 the state selects six balls randomly out of 47 numbered balls. The player selects six different numbers out of the first 47 pos- itive integers. Prizes are given for matching all six numbers (the jackpot), five numbers ($2500), four numbers ($100), and three numbers ($5.00). A ticket costs $1.00 and this dollar is not returned to the player. Find the probabil- ities of matching (a) six numbers, (b) five numbers, (c) four numbers, and (d) three numbers. (e) What is the expected value of the game to the player if the jackpot equals $1,000,000? (f) What is the expected value of the game to the player if the jackpot equals $2,000,000?