Question

In the casino game roulette, the probability of winning with a bet on red is p = 17/38. Let Y equal the number of winning bets out of 1000 independent bets that are placed. Find P(Y > 500), approximately.

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Answer 1

The probability of winning with a bet on red is p = 17/38.

Let Y equal the number of winning bets.

Y Binomial (n=1000, p=17/38)

Since n is a large number and p is close to 1/2, the binomial distrbution is approximated by normal distribution.

Y Normal (n*p = 447.3684, n*p*q = 247.2299)

By continuity correction,

P(Y > 500) = P(Y > 500.5)

                  = P(Z > (500.5 - 447.3684) / sqrt(247.2299)) ....... ( P(Z > (500.5 - mean) / sqrt(variance)))

                  = P(Z > 3.38)

                 = 1 - P(Z < 3.38)

                 = 1 - 0.99964 .......(from normal tables)

                = 0.00036

The probability that the number of winning bets are greater than 500 is approximately 0.00036.