Binomial Central Limit Theorem

[penguinnnnnx5]

Homework Statement

Here are the problems:

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.

Homework Equations

(X - np) / sqrt(np(1-p))

The Attempt at a Solution

So I've tried to implement the central limit theorem with binomial properties.

n = 1000, p = 1/38, X = 500 based on an example from the lecture slides here and here

However, when I plug everything in, everything is way too high as shown:

(500 - 1000/38) / √(1000/38 * 37 / 38) = 93.57775

Since they are so high, I cannot use this normal distribution table I was provided.

I have no idea how to do these types of problems. If anyone can please kindly explain to me the process, it would be very helpful and I will be very grateful. You don't even have to tell me the answer, or you can only do one of the questions as an example. I just want to know how it's done please.

 

[Ray Vickson]

Homework Statement

Here are the problems:

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.

Homework Equations

(X - np) / sqrt(np(1-p))

The Attempt at a Solution

So I've tried to implement the central limit theorem with binomial properties.

n = 1000, p = 1/38, X = 500 based on an example from the lecture slides here and here

However, when I plug everything in, everything is way too high as shown:

(500 - 1000/38) / √(1000/38 * 37 / 38) = 93.57775

Since they are so high, I cannot use this normal distribution table I was provided.

I have no idea how to do these types of problems. If anyone can please kindly explain to me the process, it would be very helpful and I will be very grateful. You don't even have to tell me the answer, or you can only do one of the questions as an example. I just want to know how it's done please.

1) You might as well learn the correct terminology now: the central limit theorem applies to general random variables; the name of the result applied to the binomial is the DeMoivre-Laplace limit theorem. It is more powerful than the general central limit theorem, because it can be made to give error estimates, etc.

2) Writing down probability formulas should be your last step, not your first. You need to first figure out what probability you want to calculate; only then should you worry about what formula to use. So the first question you need to answer is: what do you mean by "winning", as in the phrase "winning after 34 bets""? Say in the 34 bets you win W times and lose L = 34 - W times. In terms of W, what do you mean when you say you are winning? (I know what I would mean by that, but I want to know what you mean.) Please deal first with this question; don't write down anything else until you have answered it.

 

[penguinnnnnx5]

In terms of W, what do you mean when you say you are winning?​

To me, it seems that the problem is stating that the player is winning for N amount of times over and over, where N is the number of consecutive bets the player makes.

 

[LCKurtz]

In terms of W, what do you mean when you say you are winning?​

To me, it seems that the problem is stating that the player is winning for N amount of times over and over, where N is the number of consecutive bets the player makes.

That's so vague as to be meaningless. Ray asked you to give an answer in terms of $W$. That means an equation or inequality that $W$ must satisfy to be "winning" after 34 (or $N$) plays.

 

[penguinnnnnx5]

That's so vague as to be meaningless. Ray asked you to give an answer in terms of $W$. That means an equation or inequality that $W$ must satisfy to be "winning" after 34 (or $N$) plays.

Oh I didn't read that as closely as I should have... my apologies.

Things are starting to come together now that I've read the problem a few more times. I mistook "Winning after 34 bets" to be "winning consecutively 34 times".

I would say that $35(W) > 1(N-W)$ in order to be winning, where N is the number of times I've placed bets and W is the times I've won. This is because for every win, I receive $35 and for every loss, I lose $1. So I must win enough times to have more money than I've lost.

Am I going in the right direction now?

 

[Ray Vickson]

Oh I didn't read that as closely as I should have... my apologies.

Things are starting to come together now that I've read the problem a few more times. I mistook "Winning after 34 bets" to be "winning consecutively 34 times".

I would say that $35(W) > 1(N-W)$ in order to be winning, where N is the number of times I've placed bets and W is the times I've won. This is because for every win, I receive $35 and for every loss, I lose $1. So I must win enough times to have more money than I've lost.

Am I going in the right direction now?

Right: your net winnings = 35W - (n-W) = 36 W - n, and this should be >=0 (or maybe >=1 if say we want to go home with at least $1 more than we came with). So we either have {36 W >= n} or {36 W >= n+1}, depending on which of the two definitions of "winning" you want to use---myself, I prefer the second one.