AP Statistics 51 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

Practice

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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,

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( )