The Practice of Statistics Third Edition Chapter 8: The Binomial and Geometric Distributions Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

Binomial Distribution Examples Guessing on a multiple choice test {correct, incorrect} A basketball player shoots a free throw; the outcomes of interest are {she makes the shot, she misses} A young couple prepares for their first child {boy, girl} A quality control inspector selects an automotive part coming off the assembly line; he is interested in whether or not the part meets the requirements. It’s important to recognize a binomial setting.

Binomial Setting If data are produced in a a binomial setting, then the random variable X = number of successes is called a binomial random variable, and the probability distribution of X is called a binomial distribution. The most important skill for using binomial distributions is the ability to recognize situations to which they do and don’t apply.

Note that like the Normal distribution, the binomial distribution is a function of 2 variables.

Blood Types If both parents carry genes for O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. We are interested in the number of O blood types among five children. Is X a binomial distribution? Identify X in terms of B(n, p).

Dealing Cards Deal 10 cards from a shuffled deck and count the number X of red cards. A success is a red card. Is this a binomial distribution? sweet

Inspecting Switches An engineer chooses an SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. The engineer counts the number X of bad switches in the sample. Is this a binomial distribution? If so, identify B(n, p).

Aircraft Engine Engineers define reliability as the probability an item will perform its function. If an aircraft engine turbine has probability of performing properly for an hour of flight, the number of turbines in a fleet of 350 engines that fly for an hour without failure has what distribution?

How to calculate binomial probabilities Once we determine that a distribution is binomial, there is a formula that we can use to calculate probabilities.

How to calculate binomial probabilities The probability of having O blood type is If these parents have 5 children, what is the probability that exactly 2 will have O type blood?

How to calculate binomial probabilities The number of switches that fail inspection has approximately the binomial distribution with n = 10 and p = 0.1. The probability that no more than 1 switch fails is…

How to calculate binomial probabilities The number of switches that fail inspection has approximately the binomial distribution with n = 10 and p = 0.1. The probability that no more than 1 switch fails is… With a calculator… (2 nd /Distr). About 74% of all samples will contain no more than 1 bad switch. A sample of size 10 cannot be trusted to alert the engineer to the presence of unacceptable items in the shipment.

Rule for the union of non-disjoint events P(A or B) = P(A) +P(B) –P(AB)

Corinne’s free throws Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a key game, Corinne shoots 12 free throws and makes only 7 of them. Identify the distribution. What is the probability that she will make at most 7 free throws?

Binomcdf(12,.75, 7)

Only works with binomial distributions!

Bad Switches again B(10, 0.1) Find the mean and standard deviation of the binomial distribution.

Normal Approximation of Binomials A useful fact– as the number of trials n gets larger, the binomial distribution gets close to a Normal distribution. When n is large, we can use Normal probability calculations to approximate hard-to-calculate binomial probabilities.

Attitudes Towards Shopping A survey asked a nation-wide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time- consuming.” Suppose that in fact 60% of all adult U.S. residents would say “Agree”. What is the probability that 1520 or more of the sample agree? Independent?

Attitudes Towards Shopping A survey asked a nation-wide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time- consuming.” Suppose that in fact 60% of all adult U.S. residents would say “Agree”. What is the probability that 1520 or more of the sample agree? Find the mean and standard deviation.

Attitudes Towards Shopping A survey asked a nation-wide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time- consuming.” Suppose that in fact 60% of all adult U.S. residents would say “Agree”. What is the probability that 1520 or more of the sample agree? Use the normal calculations with this mean and standard deviation.

Attitudes Towards Shopping A survey asked a nation-wide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time- consuming.” Suppose that in fact 60% of all adult U.S. residents would say “Agree”. What is the probability that 1520 or more of the sample agree? This is very close to the software result of

As n increases, the accuracy of the Normal approximation increases. The approximation is most accurate for any fixed n when p is close to 0.5 and least accurate when p is near to 0 or 1 and the distributions are skewed.

Remember from Chapter 6: 2 Types of Simulation 1. Simulation with a Fixed Number of Trials Number of trials determined by the problem. Binomial probability 2. Wait-time Simulation We continue to simulate until a certain condition is met. Geometric probability

Binomial (Fixed Number of Trials) Simulation 1. State the problem or describe the random phenomenon. 2. State the assumptions. (individual likeliness, independent trials) 3. Assign digits to represent outcomes. Are repeated digits okay? 4. Simulate many repetitions. 5. State your conclusions.

Binomial (Fixed Number of Trials) Simulation 1. State the problem or describe the random phenomenon. 2. State the assumptions. (individual likeliness, independent trials) 3. Assign digits to represent outcomes. Are repeated digits okay? 4. Simulate many repetitions. 5. State your conclusions.

Corinne’s free throws Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a key game, Corinne shoots 12 free throws and makes only 7 of them. How unusual is it that she will make at most 7 free throws? We calculated this value to be using the binomial cdf command.

Corinne’s free throws Use your calculator to simulate this situation. 1.Let X = number of hits in 12 free-throw attempts. 12 attempts constitute 1 trial. 2.Free throws are independent and each attempt has the same likelihood of Digit 0 will represent missing a free-throw (failure). Digit 1 will represent making a free-throw (success). 4.We will use randBin(1,.75, 12) to simulate many repetitions and we will count how many free-throws are made on each of the trials. 5.We will state our conclusion.

Corinne’s free throws To automate the process a bit more, assign the results to L 1 and sum the entries of the list by using the following command: randBin(1,.75, 12) → L 1 : sum (L 1 ) (Sum is found under List then Math) The number displayed on the screen is the number of free- throws made on the current trial. Keep hitting enter until you have at least 25 trials.

Corinne’s free throws To do this simulation using a RNT: 1. Assign to miss and to make a free-throw. 2. Peel off 12 pairs of numbers from the RNT and determine whether each pair represents a missed or made free throw. 3. Determine if the trial has 7 or less free-throws. 4. Repeat steps 2 & 3 until you have an adequate number of trials. 5. Determine the proportion of trials that have 7 or less free- throws.