+ Reply to Thread
Results 1 to 9 of 9

Thread: Prob distribution

  1. #1
    Points: 81, Level: 1
    Level completed: 62%, Points required for next Level: 19

    Posts
    8
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Prob distribution




    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

  2. #2
    TS Contributor
    Points: 22,410, Level: 93
    Level completed: 6%, Points required for next Level: 940

    Posts
    3,020
    Thanks
    12
    Thanked 565 Times in 537 Posts

    Re: Prob distribution

    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.

  3. #3
    Points: 81, Level: 1
    Level completed: 62%, Points required for next Level: 19

    Posts
    8
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Prob distribution

    Quote Originally Posted by BGM View Post
    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.
    Does that means I should construct a table from $1 bet up to say 10 bets, with each bet is a double amount of the previous bet? The probability distribution is binomial as each trial - bet is independent, and identical. X~(n, 18/38)

  4. #4
    TS Contributor
    Points: 22,410, Level: 93
    Level completed: 6%, Points required for next Level: 940

    Posts
    3,020
    Thanks
    12
    Thanked 565 Times in 537 Posts

    Re: Prob distribution

    Yes you got a sequence of i.i.d. Bernoulli random variables X_i, but you are not simply summing them to got a Binomial distribution; instead you have

    (2X_1 - 1) + 2(2X_2 - 1) + 2^2(2X_3 - 1) + 2^3(2X_4 - 1) +\ldots

  5. #5
    Points: 81, Level: 1
    Level completed: 62%, Points required for next Level: 19

    Posts
    8
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Prob distribution

    Quote Originally Posted by BGM View Post
    Yes you got a sequence of i.i.d. Bernoulli random variables X_i, but you are not simply summing them to got a Binomial distribution; instead you have

    (2X_1 - 1) + 2(2X_2 - 1) + 2^2(2X_3 - 1) + 2^3(2X_4 - 1) +\ldots
    I dont get the (2X_1 - 1) part. Why do u multiple the random variable by 2 and then minus 1?

  6. #6
    TS Contributor
    Points: 22,410, Level: 93
    Level completed: 6%, Points required for next Level: 940

    Posts
    3,020
    Thanks
    12
    Thanked 565 Times in 537 Posts

    Re: Prob distribution

    It is because those X_i are Bernoulli random variables which take values \{0, 1\} only; and I use X_i = 1 to represent the event that winning on that bet (obtaining red color)

  7. #7
    Points: 81, Level: 1
    Level completed: 62%, Points required for next Level: 19

    Posts
    8
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Prob distribution

    Quote Originally Posted by BGM View Post
    It is because those X_i are Bernoulli random variables which take values \{0, 1\} only; and I use X_i = 1 to represent the event that winning on that bet (obtaining red color)
    Ok, i get it now. But it is not the probability distribution, it is just the probability equation. I'm guessing if the probability distribution is an uniform distribution as each bet has the same probability.

  8. #8
    TS Contributor
    Points: 22,410, Level: 93
    Level completed: 6%, Points required for next Level: 940

    Posts
    3,020
    Thanks
    12
    Thanked 565 Times in 537 Posts

    Re: Prob distribution

    Yes, you have correctly figure out the support of distribution.

    The outcome after n bets can be written as

    \sum_{i=1}^n 2^{i-1}(2X_i - 1) = \sum_{i=1}^n 2^iX_i - (2^n - 1)

    The summation here actually can be regarded as a binary number with n + 1 digits:

    X_nX_{n-1}\ldots X_{1}0

    So each realization of X_i will have a unique sum - and the sum will be the even integers ranging from 0, 2, \ldots, 2^{n+1} - 2.

    And you just need to state these 2^n support points after subtracting the constant. The problem here is that \Pr\{X_i = 1\} is not necessarily one-half so you may not have a discrete uniform distribution at the end.

  9. #9
    Points: 81, Level: 1
    Level completed: 62%, Points required for next Level: 19

    Posts
    8
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Prob distribution


    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?

+ Reply to Thread

           




Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts






Advertise on Talk Stats