AP Statistics51 Days until the AP Exam

The Binomial Distribution

The Binomial Coefficient

Mean and Standard Deviation

The Binomial Setting

• Each observation falls into one of two categories

• The probability of success is the same for each observation

• The observations are ALL independent

• There is a fixed number n of all observations.

The Binomial Distribution

• If data is produced in the binomial setting then the random variable X is called the binomial random variable.

• The probability distribution of X is called the binomial distribution.

• The parameters of X are n and p.• n is the number of observations• p is the probability of the observation

occurring.

The Binomial Coefficient

• This is read “n choose k”• The “!” means factorial• n is the total number of observations• k is the number of observations you are

interested in

)!(!

!

knk

n

k

n

Practice

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Practice in the Calculator

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Developing the Binomial Formula

• Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 3 children, what is the probability distribution?

X

P(X)

Outcomes Probability Rewritten

And now to introduce:

• The Binomial Probability Formula:• If X has the binomial distribution with n observations and

probability p of success on each observation, the possible values of X and 0, 1, 2, ... , n. If k is any one of these values,

knk ppk

nkXP

)1()(

Example 1:

• Twenty-five percent of the customers entering a grocery store between 5 p.m. and 7 p.m. use an express checkout. Consider five randomly selected customers, and let X denote the number among the five who use the express checkout.

• a. What is the probability that two used express checkout?

• b. What is the probability that at least four used express checkout?

Example 2

• “Do you believe your children will have a higher standard of living than you have?” This question was asked to a national sample of American adults with children in a Time/CNN poll (1/29,96). Assume that the true percentage of all American adults who believe their children will have a higher standard of living is .60. Let X represent the number who believe their children will have a higher standard of living from a random sample of 8 American adults.

• a. Interpret and find the numerical answer.• b. Find the probability that none of the parents believe

their children will have a higher standard.

The Mean and Standard Deviation of a Binomial Random Variable

• If a count X has the binomial distribution with number of observations n and probability of success p, the mean

and standard deviation of X are

• Now, as the number of trials, n, gets large, the binomial distribution becomes more normal. We can then use the normal distribution to approximate the binomial. The notation B(n,p) becomes N( )

np)1( pnp

np, np(1 p)