# Binomial Central Limit Theorem

• penguinnnnnx5
In summary: So, for example, if I win 20 times and lose 14 times, then I have won 6 times, and I would regard that as "winning" if I end up with $1 more than I came with, so I want {36*6 >= 34+1}. Now, is it a fair statement that my winnings are a binomial random variable? (I know the answer, but you tell me.)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
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.

penguinnnnnx5 said:

## 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.

Last edited:
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.

penguinnnnnx5 said:
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.

LCKurtz said:
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?

penguinnnnnx5 said:
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.

1.

## What is the Binomial Central Limit Theorem?

The Binomial Central Limit Theorem is a statistical concept that states that as the sample size increases, the distribution of sample means of a binomial distribution will approach a normal distribution. This means that even if the underlying population is not normally distributed, the sample means will approximate a normal curve.

2.

## Why is the Binomial Central Limit Theorem important?

The Binomial Central Limit Theorem is important because it allows us to make statistical inferences about a population based on a sample, even if the population is not normally distributed. This is essential in many fields of science, as most real-world data does not follow a perfect normal distribution.

3.

## How is the Binomial Central Limit Theorem applied in research?

The Binomial Central Limit Theorem is often used in hypothesis testing and confidence interval calculations. It allows researchers to make conclusions about a population based on a sample, and to determine the probability that a certain outcome is due to chance.

4.

## What are the assumptions of the Binomial Central Limit Theorem?

The main assumptions of the Binomial Central Limit Theorem are that the sample is randomly selected, the sample size is sufficiently large (usually n ≥ 30), and the observations are independent. Additionally, the Binomial Central Limit Theorem assumes that the probability of success (p) remains constant for each trial.

5.

## How does the Binomial Central Limit Theorem differ from the Central Limit Theorem?

The Central Limit Theorem applies to any population distribution, while the Binomial Central Limit Theorem only applies to binomial distributions. Additionally, the Central Limit Theorem deals with sample means, while the Binomial Central Limit Theorem deals with sample proportions. However, both theorems share the same principle that as sample size increases, the distribution of sample statistics will approach a normal distribution.

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