# Category Archives: Binomial

## Identifying a Biased Coin

You have a fair coin and a biased coin, but you can’t tell which is which. The biased coin lands on heads 75% of the time. You decide to try to determine which coin is the biased coin by selecting one of the coins at random and flipping tn 100 times. Let $$\hat{p}$$ be your observed fraction of heads. Based on $$\hat{p}$$, you decide which coin is the biased one.

1. For which values of $$\hat{p}$$ will you assume the coin you flipped is the biased coin?
2. What is the probability that you correctly determine which coin is the biased coin?

## Working Batteries

A warehouse stores batteries. Most of the batteries work properly, but about 0.1%\$ are faulty.

If a company orders 500 batteries, what is the probability that less than 3 will be faulty? Do this problem three ways:

1. Find the probability exactly.
2. Use a Poisson approximation to estimate.
3. Use a normal approximation to estimate.

A company needs 10,000 working batteries. How many batteries should the company order from the warehouse in order to be 99.7% certain that they will receive at least 10,000 working batteries?

## Biased coin

You have a biased coin, but you don’t know what the bias is.  Let $$p$$ be the actual probability of getting heads on a single coin flip, $$p=\mathbb{P}(Heads).$$

1. Suppose $$p=0.8$$. What is the probability of observing between 76 and 84 heads out of 100 flips of the coin.
2. Suppose you flip the coin 100 times and observe 80 heads. What is the 95% confidence interval for $$p$$?

## August Birthdays

About 9% of birthdays (in the US) are in August. A researcher samples 10,000 people from the US and asks for their birthdays. Estimate the probability that between 850 and 950 of those people were born in August.

## Mean and Mode of Binomial

Consider a binomial$$(10,p)$$ distribution. If $$p$$ is chosen uniformly at random from the interval $$(0,1)$$, what is the likelihood that the most likely number of the binomial distribution will be less than the mean of the binomial distribution?

## Shaq Free Throws

Over his career, Shaquille O’Neal made about 53% of his free throws. Assume his probability of making a single free throw is 53%. Suppose Shaq shot a round of 20 free throws and you’re told he made 15 of them.

1. What is the likelihood he made the first free throw, given that he made 15?
2. What is the likelihood he made at least 1 out of his first 5 free throws, given that he made 15?

You have a pair of fair dice and a pair of loaded dice. But you forgot which pair is which. You do remember that when you bought the loaded dice, the company that makes them claimed the dice would land on a sum of 7 approximately 1/3 of the time.

1. You choose one of the pairs at random and roll it once. You get a sum of 7. What is the likelihood that you picked the loaded dice?
2. You choose one of the pairs at random and roll the pair three times. You get exactly one sum of 7. What is the likelihood that you picked the loaded dice?

## Classroom Surveys

A researcher is collecting data from 10 high school classrooms. Each classroom contains 30 people. The researcher asks each student to fill out a survey. Suppose each student has about a 40% chance of completing the survey (independent of other students). What is the probability that at least 4 classrooms have at least 15 students who complete the survey?

## Defective Machines

Suppose that the probability that an item produced by a certain machine will be defective is 0.12.

1.  Find the probability (exactly)  that a sample of 10 items will contain at most 1 defective item.
2. Use the Poisson to approximate the preceding probability. Compare your two answers.

[Inspired Ross, p. 151,  example 7b ]

## Boxes without toys

A cereal company advertises a prize in every box of its cereal. In fact, only about 95% of the boxes have a prize in them. If a family buys one box of this cereal every week for a year, estimate the chance that they will collect more than 45 prizes. What assumptions are you making ?

[Pitman p122, # 9]

## Adding binomials with equal success probability

Suppose $$X \sim \text{Bin}(10,0.2)$$ and $$Y \sim \text{Bin}(5,0.2).$$

(a) Say $$X_1,\ldots,X_{10}$$ are iid with $$X = X_1 + \cdots + X_{10}$$.  What distribution for the $$X_i$$ makes this statement true?

(b) Say $$Y_1,\ldots,Y_{5}$$ are iid with $$Y = Y_1 + \cdots + Y_{5}$$.  What distribution for the $$Y_i$$ makes this statement true?

(c) Write $$X + Y$$ in terms of $$X_1,\ldots,X_{10}$$ and $$Y_1,\ldots,Y_{5}$$.

(d) What is the distribution of $$X + Y$$?

[Author Mark Huber. Licensed under Creative Commons.]

## Approximating binomial probabilities with Stirling

Let $$X$$ be a binomially distributed random variable with parameters $$n = 1950$$ and $$p = 0.342$$.

(a) Approximate $$\mathbf{P}(X = 700)$$ using Stirling’s approximation to eight significant digits.

(b) Find $$\mathbf{P}(X = 700)$$ exactly to eight significant digits using Wolfram Alpha.

[Author Mark Huber. Licensed under Creative Commons.]

## Blocks of Bernoulli Trials

In $$n+m$$ independent $$\text{Bernoulli}(p)$$ trials, let $$S_n$$ be the number of successes in the first $$n$$ trials and $$T_m$$ number of successes in the last $$m$$ trials.

1.  What is the distribution of $$S_n$$ ? Why ?
2. What is the distribution of  $$T_m$$ ? Why ?
3. What is the distribution of $$S_n+T_m$$ ? Why ?
4. Are $$S_n$$ and $$T_m$$ independent ? Why ?

[Pitman p. 159, # 10]

## Variance of binomial vs. hypergeometric

Given $$N$$ balls of which $$r$$ of them are red and the rest are green. Denote $$X$$ as the number of red balls drawn when sampling with replacement and $$Y$$ as the number of red balls drawn when sampling without replacement.

1. What is the difference between the variance of $$X$$ and the variance of $$Y$$ ?
2. For what values of $$N,n,r$$ is the variance the largest ?

## Approximation: Rare vs Typical

Let $$S$$ be the number of successes in 25 independent trials with probability $$\frac1{10}$$ of success on each trial. Let $$m$$ be the most likely value of S.

1. find $$m$$
2. find the probability that  $$\mathbf{P}(S=m)$$ correct to 3 decimal places.
3. what is the normal approximation to $$\mathbf{P}(S=m)$$  ?
4. what is the Poisson approximation to $$\mathbf{P}(S=m)$$ ?
5. repeat the first part of the question with the number of trial equal to 2500 rather than 25. Would the normal or Poisson approximation give a better approximation in this case ?
6. repeat the first part of the question with the number of trial equal to 2500 rather than 25 and the probability of success as $$\frac1{1000}$$ rather that $$\frac1{10}$$ . Would the normal or Poisson approximation give a better approximation in this case ?

[Pitman p122 # 7]

## Committee membership in the senate

A club contains 100 members; 51 are Democrats (or caucus with
Democrats) and 49 are Republicans. A committee of 10 members is
chosen at random.

1. Compute the probability of Republicans on the committee for $$n=1,…,10$$.
2. Find the probability that the committee members are all the same party.
3. Suppose you didn’t know how many Democrats there were in the senate. You observe that the committee of $$10$$ members consists of $$k=7$$ Democrats. Compute $$\mathbf{P}(M|k=7)$$, where  $$M$$ is the number of Democrats in the Senate.