10. (12%) Keno: Keno game is a game with 80 numbers 1, 2, … , 80where 20 numbered balls out of these 80 numbers will be pickedrandomly. You can pick 4, 5, 6, or 12 numbers as shown in theattached Keno payoff / odds card. When you pick 4 numbers, there isthis 4-spot special that you place $2.00 in the bet, and you arepaid $410 if all your 4 numbers are among the 20 numbers, or yourare paid $4.00 if 3 of the 4 numbers are among the 20 numberschosen from the 80 numbers. The number of ways of picking 20numbers from 80 is C(80, 20) = 80! / (60! 20!). The number of waysthat all your 4 numbers are among the 20 numbers is: C(76, 16)(why?) = 76! / (60! 16!). The probability that your 4 numbers bingois C(76, 16) / C(80, 20), and the theoretical payoff should beC(80, 20) / C(76, 16) = 80! * 16! / (76! * 20!) = (80 * 79 * 78 *77 ) / (20 * 19 * 18 * 17 ) = $326.4355… (a) (6%) Based on thiscomputation, is the payoff fair? Explain! (b) (6%) The payoff for 3numbers in your 4 chosen numbers appear in the 20 numbers is $4.00.Is that a fair payoff (how is the number of ways of 3 numbersmatching related to the number of ways of 4 numbers matching?)?
