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Math Help - roulette win probability

  1. #1
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    roulette win probability

    In the game of roulette the probability of winning is 18/37. Suppose that in each game a player earns 1 lira or loses 1 lira.


    (a) (5 pts) How many games must be played in a casino in order that the casino earns, with probability 1/2, at least 1000 liras daily?


    (b) (5 pts) Find the percentage of days on which the casino has a loss.
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  2. #2
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    (a) The earning of each game can be modeled as a Bernoulli random variable with mean 1/37 and variance 1368/1369. If N games are played, then the casino earnings are the sum of N iid Bernoulli random variable, which has a binomial distribution, or since N is large, by Central Limit Theorem, it is approximately normal.

    So if X denotes the casino earnings, then P(X > 1000) = 0.5 means, by symmetry of the normal distribution, that X has a mean of 1000. Since the Bernoulli random variable has a mean of 1/37, the casino needs to play 37000 games.

    (b) Assuming that N games are played and N is large enough such that CLT is applicable, then
    P(X < 0) = P(\dfrac{X-\dfrac{N}{37}}{\sqrt{\frac{1368N}{1369}}} < \dfrac{0-\dfrac{N}{37}}{\sqrt{\frac{1368N}{1369}}}) = P(Z < \dfrac{0-\dfrac{N}{37}}{\sqrt{\frac{1368N}{1369}}})
    where Z is standard normal
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  3. #3
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    i wanna see the calculations.

    how can u find 1/37 and variance 1368/1369. can u be more clear in part of a. im trying to understand. anyway, thank u.
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