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Section 6.3
Binomial Distributions
A Gaggle of Girls
Let’s use simulation to find the probability
that a couple who has three children has
all girls.
P(girl) = 0.5
Let 0 = boy and 1 = girl.
Use your calculator to choose 3 random
digits to simulate this experiment.
Complete this experiment 50 times in your
group and record. Create a probability
distribution for X = number of girls.
Gaggle continued
What was your group’s probability for
having three girls?
Use your knowledge of probabilities to find
the actual chance that a family with three
children has three girls.
Are these close?
Children, Again???
Two types of scenarios:

A couple is going to have children until they
have a girl.
Here, the random variable is how many children
will it take to get a girl.

A couple is going to have 3 children and we’ll
count how many are girls.
Here, the random variable is how many girls there
are out of the 3 children.
Dichotomous Outcomes
Both of those situations have dichotomous
(two) outcomes.
Other examples with two outcomes:



Coin toss (heads or tails)
Shooting free throws (make or miss)
A game of baseball (win or lose)
Special Type of Setting
In this chapter, we’ll study a setting with
two outcomes where there are a fixed
number of observations (or trials).
The binomial distribution is a special
type of setting in which there are two
outcomes of interest.
4 Conditions for a Binomial Setting
1. There are two outcomes for each
observation, which we call “success” or
“failure.”
2. There is a fixed number n of
observations.
3. The n observations are all independent.
4. The probability of success, called p, is
the same for each observation.
Binomial Random Variables
Binomial random variable: In a binomial setting,
the random variable X = # of success.
The probability distribution of X is called a
binomial distribution.

The parameters of a binomial distribution are n (the
number of observations) and p (the probability of
success on any one observation).
B(n, p)
Is a binomial random variable discrete or
continuous?
Discrete…
Example
Blood type is inherited. If both parents
have the genes for the O and A blood
types, then each child has probability 0.25
of getting two O genes and thus having
type O blood. Is the number of O blood
types among this couple’s 5 children a
binomial distribution?


If so, what are n and p?
If not, why not?
Example
Deal 10 cards from a well-shuffled deck of
cards. Let X = the number of red cards. Is
this a binomial distribution?


If so, what are n and p?
If not, why not?
Using the Calculator to Find
Binomial Probabilities
Under 2nd VARS (DISTR), find 0:binompdf(
This command finds probabilities for the
binomial probability distribution function.
The parameters for this command are
binomialpdf(n, p, x) IN THAT ORDER.

This will only give you the probability of a
single x value.
Example
Let’s go back to the couple having three
children. Let X = the number of girls.
p = P(success) = P(girl) = 0.5
The possible values for X is 0, 1, 2, 3.
Using the binompdf(n,p,x) command,
complete the probability distribution.
What is the probability that the couple will
have no more than 1 girl?
Cumulative Distribution Function
The pdf command lets you find
probabilities for ONE value of X at a time.
binomialcdf(n, p, x)

This time, you will be given the sum of the
probabilities ≤ x. Be sure you remember this
when answering a question
The cdf command finds cumulative
probabilities. We can use it to quickly find
probabilities such as P(X < 7) or P(X ≥ 4).
Corinne’s Free Throws
Corinne makes 75% of her free throws
over the course of a season. In a key
game, she shoots 12 free throws and
makes 7 of them. Is it unusual for her to
shoot this poorly or worse?
What is the probability that Corinne makes
at least 6 of the 12 free throws?
Homework
Chapter 6#
69-72, 86, 94