Question in the picture. Introduction to Stochastic Process. MATH 447 in McGill.4.26 In a lottery game, three winning numbers are chosen uniformly at random from {15, 100}, sampling without replacement. Lottery tickets cost $1 and allow a player to pick three numbers.
If a player matches the three winning numbers they win the jackpot prize of $1,000. For matching exactly two numbers, they win $15. For matching exactly one number they win $3. (3) Find the distribution of net winnings for a random lottery ticket. Show that the expected value of the game is 70.8 cents.
(b) Parlaying bets in a lottery game occurs when the winnings on a lottery (C) ticket are used to buy tickets for future games. Hoppe (2007) analyzes the effect of parlaying bets on several lottery games. Assume that if a player matches either one or two numbers they parlay their bets, buying respectively 3 or 15 tickets for the next game. The number of tickets obtained by parlaying can be considered a branching process. Find the mean of the offspring distribution and show that the process is subcritical. See Exercise 4.19. Let T denote the duration of the process, that is, the length of the parlay. Find P(T = k), for k = l, ,4.
(d) Hoppe shows that the probability that a single parlayed ticket will ultimately win the jackpot is approximately p / (l m), where p is the probability that a single ticket wins the jackpot, and m is the mean of the off spring distribution of the associated branching process. Find this probability and Show that the parlaying strategy increases the probability that a ticket will ultimately win the jackpot by slightly over 40%.