I suspect the "financial outcome" refers to the outcome after a certain number of bets (not the first bet only); but it seems that it does not mention up to which bet.
The roulette wheel has 38 “pockets”.
The “pockets” are numbered 1, 2, …, 36; 0 and 00.
Half of the “pockets” 1, 2, …, 36 are coloured red, the other half are black. 0 and 00 are green.
If a player outlays $b in a bet on red, and the result of the wheel’s spin is
* red, then the casino (“house”) pays the player $2b (ie, the $b outlaid in the bet, plus $b).
* not red, then the casino keeps the $b that the player bet.
Felipe’s strategy was:
Bet #1: $1 on red.
Bet #2: $2 on red.
Bet #3: $4 on red. etc
Assume that the wheel is “fair”, that is, a ball thrown into the spinning wheel is equally likely to fall into any one of the 38 “pockets”. Assume that the casino does not impose betting limits.
(a) Consider one spin of the wheel, with a player betting $1 on red.
(i) What is the probability that the player will win?
Explain how you obtain your answer and round it to 6 decimal places.
(ii) Tabulate the probability distribution of the player’s financial outcome.
Explain how you obtain your answer.
(a) (i) is that P(W) = 18/38 = 0.473684211. It is the probability to draw a red pocket from total 38 pocket, where the red pocket contains 18 out of total 38 pockets in the wheel.
(ii) but I'm not sure how to tabulate the probability distribution of the player's financial outcome. It is asking only 1 spin and placing 1 bet. Isn't it only one outcome whether win or not win. The financial player outcome will be either 2 or -1?
Is there anything i misunderstand the part ii question?
Thanks a lot in advance
I suspect the "financial outcome" refers to the outcome after a certain number of bets (not the first bet only); but it seems that it does not mention up to which bet.
Yes, you have correctly figure out the support of distribution.
The outcome after bets can be written as
The summation here actually can be regarded as a binary number with digits:
So each realization of will have a unique sum - and the sum will be the even integers ranging from .
And you just need to state these support points after subtracting the constant. The problem here is that is not necessarily one-half so you may not have a discrete uniform distribution at the end.
umm.. but how could i tabulate the probability distribution? I will need to create a table for the probability of each bet..
so.. does it begin with something like below?
Bet Prob Financial outcome
1 0.4737 2(0.4737)-1
2 0.4737 2[2(0.4737)-1] etc?
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