After visiting Las Vegas over spring break, our entrepreneurial engineering student abandons his

cargo hauling business and decides to take up gambling. Roulette is his game of choice because he

figures that if he bets on red every spin then he has a 0.5 probability of winning. What he doesn't

realize is that probability of landing on red each spin is 18/38, not 0.5.

His plan is to bet 1 token(chip) each spin. If the wheel lands on red, he keeps the token he has bet

and wins another one. If it doesn't land on red, he loses the token he has bet. Suppose he starts

with 3 tokens. He will keep playing until he either doubles his tokens (i.e., reaches 6 tokens total)

or goes broke, at which time he will leave the game.

1. Model the student's roulette experience as a terminating Markov chain. Use all positive funds

balances as states, and add an additional state for going broke. Remember that absorbing

states are typically given the highest state numbers and are grouped together. Provide state

definitions and draw the transition diagram.

2. How many spins can the student expect to play before he leaves the game?

3. What is the probability that the student reaches 5 chip at some point in the game?

4. How many times can the student expect to have 2 chips as his funds balance?

5. What is the probability that he leaves the game by going broke?

6. Suppose now that the student finds a game that uses a European roulette wheel, in which

case the probability of landing on red is 18/37. He ups the stakes by starting with 5 chips

but uses the same strategy of betting 1 chip on each spin. What is the probability that he

leaves the game broke if he starts with 5 chips?

7. How many spins can the student expect to play before he leaves the game in case of the

previous subproblem?