Section 8.1 ~ 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 _____________________________________________________ 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)
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)
Understanding the Binomial Formula

To best understand the formula for a binomial distribution, lets use the example of inheriting
blood type with the B(5, .25) distribution.

Each child of a particular pair of parents has probability 0.25 of having type O blood.
Genetics says that children receive genes from each of their parents independently. If
these parents have 5 children, the count X of children with type O blood is a binomial
random variable with n = 5 trials and probability p = 0.25 of a success on each trial. In
this setting, a child with type O blood is a “success” (S) and a child with another blood
type is a “failure” (F). What’s P(X = 2)?
2
3
P(SSFFF) = (0.25)(0.25)(0.75)(0.75)(0.75) = (0.25) (0.75) = 0.02637
However, there are a number of different arrangements in which 2 out of the 5
children have type O blood:
Verify that in each arrangement, P(X = 2) = (0.25)2(0.75)3 = 0.02637
Therefore, P(X = 2) = 10(0.25)2(0.75)3 = 0.2637
Binomial Coefficient

Note, in the previous example, any one arrangement of 2 S’s and 3 F’s had the same
probability. This is true because no matter what arrangement, we’d multiply together
0.25 twice and 0.75 three times.

We can generalize this for any setting in which we are interested in k successes in n
trials. That is,
Binomial Probability
 The binomial coefficient counts the number of different ways in which ______________
_____________________________________________________________.
 The binomial probability _____________ is this count ____________________________
______________________________________________________.
Example 8.10 – Defective Switches

The number X of switches that fail inspection in example 8.3 has approximately the
binomial distribution with n = 10 and p = .1. Using the binomial probability formula,
determine the probability that no more than 1 switch fails.
𝑃(𝑋 ≤ 1) = 𝑃(𝑋 = 1) + 𝑃(𝑋 = 0)
10
10
( ) (.1)1 (.9)9 + ( ) (.1)0 (.9)10
1
0
10!
10!
(. 1)(. 3874) +
(1). 3487)
1! 9!
0! 10!
(10)(. 1)(. 3874) + (1)(1)(. 3487)
. 7361