I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out…
A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet
1 on a specified number, you either win 35 if the roulette ball lands on that
number or lose 1 if it does not. If you continually make such bets, approximate the
probability that
(a) you are winning after 34 bets;
(b) you are winning after 1,000 bets;
(c) you are winning after 100,000 bets.
So i get the idea is to figure out the mean and variance so that I can let X be estimated by a normal distribution, I got the correct values and checked them with the solutions, but theres one thing I'm confused about.
Even though the solution calculates the expected value of the winnings E(W) = (35)(1/38) + (-1)(37/38) = -0.0526, it uses a different value for the mean in the calculation of the Z score.
It used 1 – (37/38)^34 = 0.596 ..
This is different than any other problem I've seen before, I was just hoping someone could explain why you use this value as opposed to the expected value E(W), and what exactly this value means.
Thanks.
Best Answer
$\left(\dfrac{37}{38}\right)^{34}$ is the probability of losing $34$ bets in a row.
If this does not happen then you have gained $35$ at least once and so are ahead after $34$ bets.
So this is not a $Z$ score but, after subtraction from $1$, an exact probability answer to (a). You do not need to consider expectations or approximations.