When playing roulette, there are 38 slots on the wheel, all equallylikely on any

Question

When playing roulette, there are 38 slots on the wheel, all equallylikely
on any spin of the wheel. The slots are numbered 1-36 plus 0 and00. One type of bet is
to wager $1 on a group of 3 numbers. If one of those numbers comesup, you get your $1
back plus $11 more, so your net gain is $11. If none of your 3numbers comes up, you
lose your $1, so your net gain is -$1.
You make this particular bet a total of n=3,600 times. Let Xidenote your net gain on the
ith play, so Xi = 11 with probability p = 3/38 and Xi = -1 withprobability 35/38. Also,
E(Xi) = = -1/19 and Var(Xi) = 2 = 3780/361 =10.4709.
Use the CLT to estimate the chance that you are ahead (i.e, yournet winnings are 0 or
greater) after your 3,600 plays.

Explanation / Answer

The mean is -1/19 and the standard deviation is 10.4709 (these weregiven to you). By the central limit theorem, the samples of 3600plays will have a mean of -1/19 and a standard deviation of(10.4709)/(3600)=.0539. This gives a z-score for"being ahead" as 0-(-1/19)/(.0539)=.9759. The probability of seeingthis z-score or higher (from a table) is .164557, or about16.456%

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