Question

ROULETTE is a casino game where a numbered wheel spins and a steel ball falls into a location marked by one particular colored number. In the United States there are 18 locations colored red, 18 locations colored black and 2 locations colored green. The red and black locations are numbered 1-36 and the green locations are labeled "0" and "00" as shown in the picture to the right. The wheel therefore has 38 locations in total. Note that the odd and even values are not evenly distributed within each color. For any particular wager a player makes, an expected profit can be calculated from: Expected profit = (Prob. of winning) x (winning payout amount) - wager

(b) To “let it ride” is to make a bet and if it wins to make the same bet with the entire payout amount. For example, if you bet $1 on black and win the casino gives you $2, you then "let it ride" by betting that $2 on black and if you win again the casino would give you $4. If the ball were to land on red or green on either spin of the wheel you would lose your wager and the final payout would be $0. What is the expected profit for betting $15 on black and if you win, "letting it ride" once? I.E., betting on black and if you win betting your winnings on black again? (keep in mind that if you lose the first bet you are done; round to closest penny)

Answer 1

Solution:-

Given that

There are 18 locations colored red, 18 locations colored black and 2 locations colored green. The red and black locations are numbered 1-36 and the green locations are labeled "0" and "00" as shown in the picture to the right. The wheel therefore has 38 locations in total. Note that the odd and even values are not evenly distributed within each color. For any particular wager a player makes, an expected profit can be calculated from:

Expected profit = (Prob. of winning) x (winning payout amount) - wager

There are 18 black locations out of a total of 38 locations were the ball can land.

The probability that the ball lands on a black location is

(b) What is the expected profit for betting $15 on black and if you win, "letting it ride" once? I.E., betting on black and if you win betting your winnings on black again? (keep in mind that if you lose the first bet you are done; round to closest penny)

You win $ 15 * 2 * 2 = $ 60, if you bet $15 on black and let it ride. To win, the ball needs to land of black in both the first and the second spins. The probability of ball landing on black on any given spin is 0.4737 (from part a). The probability of ball landing on black on both the spins is

P(black on 1st black on 2nd) = P(black on 1st) P(black on 2nd)

The events are independent

= 0.4737 0.4737

= 0.2244

The probability of winning is 0.2244

winning payout amount = $ 60

wager = $ 15

The expected profit is

expected profit = (Probability of winning) (winning payout amount) - wager

= 0.2244 60 - 15

= -1.536

Ans:-

The expected profit for betting $ 15 on black and if you win, letting it ride once is $ -1.54

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