1. Chapter 6: Random Variables
Section 6.3
Binomial and Geometric Random Variables
The Practice of Statistics, 4th edition – For AP*
STARNES, YATES, MOORE

2. Quick Review
What is the difference between a discrete RV and a continuous RV?
What is the table called that shows the outcomes of a discrete RV and the respective probabilities?
How do you graph a discrete RV?
Is the normal RV discrete or continuous?

3. Special Kinds of Discrete RVs
You need to train yourself to recognize the difference between these 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.
Both of these situations have binary outcomes. Others include coin tosses, shooting free throws, etc.

4. Binomial and Geometric Random Variables
Definition:
A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs. The four conditions for a binomial setting are
Binomial and Geometric Random Variables
• Binary? The possible outcomes of each trial can be classified as “success” or “failure.” B
• Independent? Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. I
• Number? The number of trials n of the chance process must be fixed in advance. N
• Success? On each trial, the probability p of success must be the same. S

5. Example
Blood type is inherited. If both parents in a couple 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?

6. Today
Now, we’ll learn about the basis for the binomial calculations: the binomial formula.
p stands for the probability of success. n represents the number of observations. k is the value of x of which you’re asked to find the probability.
Notice the = mark.
This is a combinatorial. It is read “n choose k.”

7. Combinatorials
A combination (or combinatorial) helps us find out how many ways there are to choose k objects from n total objects.
For example, let’s revisit the blood type example. If we’re trying to find the probability that a couple has 2 type O children out of 5, then the coefficient of the binomial formula is
= 5!
2!3! n!
k!(n-k)!

8. Completing the Formula
In the previous example, we were trying to find out the probability that a couple who has 5 children has 2 type 0 blood children.
The formula, then, is

9. Example: Inheriting Blood Type
Each child of a particular pair of parents has probability 0.25 of having blood type O. Suppose the parents have 5 children
(a) Find the probability that exactly 3 of the children have type O blood.
Let X = the number of children with type O blood. We know X has a binomial distribution with n = 5 and p = 0.25.
(b) Should the parents be surprised if more than 3 of their children have type O blood?
To answer this, we need to find P(X > 3).
Since there is only a 1.5% chance that more than 3 children out of 5 would have Type O blood, the parents should be surprised!

10. Try This…
The number of switches that fail inspection follows a binomial distribution with n = 10 and p = Find the probability that no more than 1 switch fails. Hint: 0! = 1.