Normal approximation to the Binomial distribution. A roulette wheel has 38 slots

Question

Normal approximation to the Binomial distribution. A roulette wheel has 38 slots numbered 0, 00, 1, 2, . . . ,36. The slots numbered 0 and 00 are green. Of the other slots, 1/2 are red and 1/2 are black. Before starting to play, you want to look for biases that you can exploit. You therefore watch 100 rounds and count the number of rounds for which the result is odd. If the count exceeds 55, you decide that the roulette is not fair. Assuming that the roulette is fair, find an approximation for the probability that you will make the wrong decision.

Explanation / Answer

no of odd numbers in wheel is 18

Probablity of odd numbers in wheel is =18/38 = 0.474

Mean no of odd numbers = 100*(0.474) = 47.4

Variance of nof odd numbers = 100*0.474*(1-0.474) = 24.93

standard deviation of odd numbers = 4.993

z value for 55 is (55-47.4)/4.993 = 1.52213 corresponding p value using z table is 0.936

P(no of odd numbers <= 55) = 0.936

P(no of odd number excedd 55) = 1-0.936 = 0.064

probability that you will make the wrong decisiont when the roulette is fair = 0.064

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