11. (14.28) A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slot

Question

11. (14.28) A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. You decide to play roulette 280 times, each time betting the same amount on red. Check that the Normal approximation is permissible, and use it to find the probability (±±0.0001) that you will lose money:

Explanation / Answer

P(losing money) = p = 1 - P(betting on red)
= 1 - 18/38
= 20/38
= 0.53
n = 280

Normal approximation to Binomial distribution is used.

Mean = np = 280*0.53 = 148.4

Standard deviation = sqrt (np(1-p))
= sqrt (280*0.53*0.47)
= 8.35

z = (X - Mean)/SD
0.0001 probability of losing money is represented by the 0.0001 area in the extreme left tail of the normal curve.

The z value which separates this area from the remaining is - 3.70

Therefore,
- 3.70 = (X - 148.4) / 9.98
- 36.926 = X - 148.4
X = 148.4 - 36.926

= 111.474 or 112 (on rounding)

z value is - 3.70 is located from the z-scores chart.
The area under the standard normal curve corresponding to z = 3.70 is 0.4999
Total area on the left side of mean = 0.5000
Therefore the area in the extreme left tail of the normal curve = 0.5000 - 0.4999 = 0.0001
The z value being on the left side of mean , it should be taken as -ve
The process followed for locating the z value is reverse to that of the procedure followed for determining the area corresponding to a specific z value.
i.e., the z value corresponding to (0.5000-0.0001) 0.4999 is to be located

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