Section 8.1 Part 1 ~ The Binomial Distributions
In practice, we frequently encounter experimental situations where there are two outcomes of interest.
Some examples are:

We use a coin toss to answer a question.

A basketball player shoots a free throw.

A young couple prepares for their first child.
The Binomial Setting
1. Each observation falls into _____________________________________, which for convenience
we call ________________________________________.
2. There is a _______________________________________ of observations.
3. The n observations are all ______________________________. (That is, knowing the results of
one observation tells you nothing about the other observations).
4. The probability of success, call it p, is ________________________
The Binomial Distribution

The distribution of _____________________________________________________is the
_________________________________________ with parameters n and p.

The parameter n is the _______________________________________, and p is the
___________________________________________________________on any one observation.

The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say that
__________________________________________________.
Example 8.1 -Blood Types

Blood type is inherited. If both parents carry genes for the O and A blood types, each child has
probability of 0.25 of getting two O genes and so of having blood type O. Suppose there are 5
children and that the children inherit independently of each other.

Is this a binomial setting? If so, find n, p and X.

n = 5, p = .25, X = B(5, .25)
Example 8.2 – Dealing Cards

Deal 10 cards from a shuffled deck and count the number X of red cards. There are 10
observations and each are either a red or a black card.

Is this a binomial distribution?


No because each card chosen after the first is dependent on the previous pick
If so what are the variables n, p and X?


None
Example 8.3 – Inspecting Switches

An engineer chooses an SRS of 10 switches from a shipment of 10,000 switches Suppose that
(unknown to the engineer) 10% of the switches in the shipment are bad. The engineer counts
the number X of bad switches in the sample.

Is this a binomial situation? Justify your answer.

While each switch removed will change the proportion, it has very little effect
since the shipment is so large.


In this case the distribution of X is very close to the binomial distribution B(10, .1)
The sampling distribution of a count variable is only well-described by the binomial distribution is cases
where the population size is _____________________________ than the sample size.

As a general rule, the binomial distribution should not be applied to observations from a simple
random sample (SRS) unless the __________________________________________________
(or otherwise thought of as the sample size being no more than 10% of the population)

𝑁 ≥ 10𝑛 or 𝑛 ≤
1
𝑁
10
Example 8.5 – Inspecting Switches

An engineer chooses an SRS of 10 switches from a shipment of 10,000 switches. Suppose that
(unknown to the engineer) 10% of the switches in the shipment are bad. What is the probability
that no more than 1 of the 10 switches in the sample fail inspection?

See explanation/diagram on p.442
pdf ~ “Probability Distribution Function”

Given a discrete random variable X, the probability distribution function assigns a probability to
_____________________________________.

The probability must satisfy the rules for probabilities given in Chapter 6.

The TI-83 command _____________________________________ will perform the calculations.

This is found under 2nd/DISTR/0
Example 8.6 – Corinne’s Free Throws

Corinne is a basketball player who makes 75% of her free throws over the course of a season. In
a key game, Corinne shoots 12 free throws and makes only 7 of them. The fans think that she
failed because she is nervous. Is it unusual for Corinne to perform this poorly?

Assume that the free throws are independent of each other.

The number X of baskets in 12 attempts has the B(12, .75) distribution.

We want the probability of making a basket on at most 7 free throws:

𝑃(𝑋 ≤ 7) = 𝑃(𝑋 = 0) + 𝑃(𝑋 = 1) + ⋯ 𝑃(𝑋 = 7)

𝑃(𝑋 ≤ 7) = .157Corinne will make at most 7 of her 12 free throws about 16%
of the time.
Example 8.7 – Three Girls

Determine the probability that all 3 children in a family are girls.

Takes on the B(3, .5) distribution

𝑃(𝑋 = 3) = binomPdf(3, . 5, 3) = .125
Cdf ~ “Cumulative Distribution Function”

The cumulative binomial probability is useful in a situation of a __________________________.

Given a random variable X, the cumulative distribution function (cdf) of X calculates the
______________________________________________________________________________

That is, it calculates the probability of obtaining _________________________in n trials.
binomPdf vs binomCdf

See example 8.8 on p.444 to see how binomPdf and binomCdf distributions compare

binomCdf is also useful for calculating the probability that it takes _________________ a certain
number of trials to see the first success.

The calculation uses the complement rule:

𝑃(𝑋 > 𝑛) = 1 − 𝑃(𝑋 ≤ 𝑛)

n = 2, 3, 4, …
Using Pdf & Cdf to Find Probabilities

Use the B(12, .75) distribution, find the following probabilities:

𝑃(𝑋 = 4)


𝑃(𝑋 ≤ 4)


cdf(3)
𝑃(𝑋 > 4)


cdf(4)
𝑃(𝑋 < 4)


pdf(4)
1-cdf(4)
𝑃(𝑋 ≥ 4)

1-cdf(3)