Each child born to a particular set of parents has probability of 0.25 having blood type O. Suppose these parents have 5 children. Let X = number of children who have type O blood. Then X is B(5, 0.25). What is the probability that exactly 2 of the

children have type O blood? Make a table for the pdf of the random variable X.

Then use the calculator to find the probabilities of all possible values X, and complete the table.

Verify that the sum of the probabilities is 1. Construct a histogram(in calc) of the pdf.

Section 6.4 Second Day

The Binomial Formula

Previously… What are the four conditions for a binomial

setting? Two outcomes (success or failure) for each

observation. P(success) is the same for each observation. There are a fixed number of observations. The observations are all independent.

What is the difference between the cdf command and the pdf command?

Today

Now, we’ll learn about the basis for the binomial calculations: the binomial formula.

( ) (1 )k n knP X k p p

k

Notice the = mark.

This is a combinatorial. It is read “n choose k.”

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.

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 “gaggle of girls” example. If we’re trying to find the probability that a couple has 1 girl out of 3 children, then the coefficient of the binomial formula is .

= 3! 1!2! *You can use nCr on your

calculator*

3

1

3

1

)!(!

!

knk

n

k

n

Completing the Formula

In the previous example, we were trying to find out the probability that a couple who has three children has one girl.

The formula, then, is

1 2

( ) (1 )

3( 1) (0.5) (1 0.5)

1

k n knP X k p p

k

P X

Try This…

The number of switches that fail inspection follows a binomial distribution with n = 10 and p = 0.1. Find the probability that no more than 1 switch fails.

Why can’t I just use binomialpdf or binomialcdf ??? That is fine for multiple choice or to check

your answer. On free response, judges expect you to at

least be able to fill in the formula. Find it on your formula sheet.

The Mean and Standard Deviation of a Binomial Random Variable

np

)1( pnp

These are the formulas for a binomial distribution ONLY. Be sure you are looking at a binomial random distribution before you make the calculations.

These formulas are on your formula sheet. Let’s locate them.

Example

If there are 10 multiple choice questions on a test, and each question has 4 answer choices, how many on average will a student get right by purely guessing? What is the standard deviation?

Note: This is much easier to use since we could have hundreds of observations to look at. It would not be advised to spend the time making your own probability distribution.

The Normal Approximation

When n is very large, the calculations for the binomial distribution become cumbersome.

In addition, as n gets large, the binomial distribution gets closer to a normal distribution. Therefore, we will use a normal approximation for

the binomial curve if certain conditions are met.

The Conditions

The conditions that must be met in order to use a normal curve to approximate the binomials are: np ≥ 10

AND n(1-p) ≥ 10

When these conditions are met, the distribution of X is approximately normal with mean np and standard deviation .)1( pnp

Example

Sample surveys show fewer people enjoy shopping than in the past. A recent survey asked a nationwide random sample of 2500 adults if they agreed or disagreed with the statement, “I like buying new clothes, but shopping is often frustrating and time-consuming.” The population that the poll wants to draw conclusions about is all U.S. residents aged 18 and over. Suppose that in fact 60% of all adult residents would say “Agree” if asked the same question. What is the probability that 1520 or more of the sample agree?

Simulations

Again??? Yes! You can simulate a binomial distribution.

Remember to assign digits based on the probability assigned to each outcome. For example, let’s say I’m going to choose 10 CMHS

students at random and see how many live outside of Concord. Suppose 40% of CMHS’s population lives outside of Concord. What is the probability that 6 or more students in my sample live outside of Concord?

Complete the Simulation As a group, complete 20 simulations of the

last problem and record how many times you get 0-10 successes. Make a probability distribution.

Calculate the actual binomial probabilities and write a sentence to compare your simulation to your actual binomial distribution.

Homework

Chapter 6

79, 81, 82, 85, 91