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Posted on Jun 28, 2024

Two gamblers play a version of roulette with a wheel as shown in the file P05_63.xlsx. Each gambler places four bets, but their strategies are

Two gamblers play a version of roulette with a wheel as shown in the file P05_63.xlsx. Each gambler places four bets, but their strategies are different, as explained below. For each gambler, use the rules of probability to find the distribution of their net winnings after four bets. Then find the mean and standard deviation of their net winnings. The file gets you started.

Round your answers to two decimal places, if necessary and if your answer is negative value, please enter "minus" sign.

a. Player 1 always bets on red. On each bet, he either wins or loses what he bets. His first bet is for $10. From then on, he bets $10 following a win, and he doubles his bet after a loss. (This is called a martingale strategy and is used frequently at casinos.) For example, if he spins red, red, not red, and not red, his bets are for $10, $10, $10, and $20, and he has a net loss of $10. Or if he spins not red, not red, not red, and red, then his bets are for $10, $20, $40, and $80, and he has a net gain of $10.

Mean	$
Standard Deviation	$

b. Player 2 always bets on black and green. On each bet, he places $10 on black and $2 on green. If red occurs, he loses all $12. If black occurs, he wins a net $8 ($10 gain on black, $2 loss on green). If green occurs, he wins a net $50 ($10 loss on black, $60 gain on green).

Mean	$
Standard Deviation	$

Below, you see a European roulette wheel, with the numbers 0 through 36 around the wheel. There are two players we will compare.

Player 1 always bets on red. He places four bets. On each bet, he either wins or loses what he bets. His first bet is for $10. From then on, he bets $10 following a win, or he doubles his bet after a loss. For example, if he spins red, red, not red, and not red, his bets are for $10, $10, $10, and $20, and he ends up with a net loss of $10. Or if he spins not red, not red, not red, and red, then his bets are for $10, $20, $40, and $80, and he ends up with a net gain of $10.

Player 2 always bets on black and green. He also places four bets. On each bet, he places $10 on black and $2 on green. If red occurs, he loses all $12. If black occurs, he wins a net $8 ($10 gain on black, $2 loss on green). If green occurs, he wins a net $50 ($10 loss on black, $60 gain on green).

We are interested in the probability distribution of net win/loss for each player separately. Would you rather be player 1 or player 2? Or would you rather not play!

Basic probabilities

P(red)

18/37

P(not red)

19/37

Possible scenarios for player 1 (order matters)

Spin 1

Spin 2

Spin 3

Spin 4

P(Spin 1)

P(Spin 2)

P(Spin 3)

P(Spin 4)

Bet 1

Bet 2

Bet 3

Bet 4

Net win/loss

Probability

player 2

Basic probabilities
P(red)	18/37
P(black)	18/37
P(green)	1/37

Net win/loss for each possible outcome
Red	Black	Green
-12	8	50

Possible scenarios for player 2 (order of spins doesn't matter)
# reds	# blacks	# greens	# combinations	Net win/loss	Probability
4	0	0
3	1	0
3	0	1
2	2	0
2	1	1
2	0	2
1	3	0
1	2	1
1	1	2
1	0	3
0	4	0
0	3	1
0	2	2
0	1	3
0	0	4