Binomial Distribution The binomial distribution is a discrete distribution.
© 2010 Pearson Prentice Hall. All rights reserved Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment if 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure. 4. The probability of success is fixed for each trial of the experiment. © 2010 Pearson Prentice Hall. All rights reserved
Binomial Experiment A binomial experiment has the following properties: experiment consists of n identical and independent trials each trial results in one of two outcomes: success or failure P(success) = p P(failure) = q = 1 - p for all trials The random variable of interest, X, is the number of successes in the n trials. X has a binomial distribution with parameters n and p
Notation Used in the Binomial Probability Distribution There are n independent trials of the experiment. Let p denote the probability of success so that 1 – p is the probability of failure. Let X be a binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 < x < n. © 2010 Pearson Prentice Hall. All rights reserved
EXAMPLES A coin is flipped 10 times. Success = head. X = n = p = Twelve pregnant women selected at random, take a home pregnancy test. Success = test says pregnant. X = n = p = ? Random guessing on a multiple choice exam. 25 questions. 4 answers per question. Success = right answer.
Examples when assumptions do not hold Basketball player shoots ten free throws Feedback affects independence and constant p Barrel contains 3 red apples and 4 green apples; select 4 apples without replacement; X = # of red apples. Without replacement implies dependence
What is P(x) for binomial?
Mean and Standard Deviation The mean (expected value) of a binomial random variable is The standard deviation of a binomial random variable is
Example Random Guessing; n = 100 questions. Expected Value = Probability of correct guess; p = 1/4 Probability of wrong guess; q = 3/4 Expected Value = On average, you will get 25 right. Standard Deviation =
Example Cancer Treatment; n = 20 patients Probability of successful treatments; p = 0.7 Probability of no success; q = ? Calculate the mean and standard deviation.
© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Identifying Binomial Experiments Which of the following are binomial experiments? A player rolls a pair of fair die 10 times. The number X of 7’s rolled is recorded. (b) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly selects flights until she finds 10 that were not on time. The number of flights X that need to be selected is recorded. (c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded. Binomial experiment Not a binomial experiment – not a fixed number of trials. Not a binomial experiment – the trials are not independent. © 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Constructing a Binomial Probability Distribution According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage of 79.0% in May, 2008. Suppose that 4 flights are randomly selected from May, 2008 and the number of on-time flights X is recorded. Construct a probability distribution for the random variable X using a tree diagram. © 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Using the Binomial Probability Distribution Function According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a random sample of 20 car-owning households, what is the probability that exactly 5 have three or more cars? (b) In a random sample of 20 car-owning households, what is the probability that less than 4 have three or more cars? (c) In a random sample of 20 car-owning households, what is the probability that at least 4 have three or more cars? © 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Finding the Mean and Standard Deviation of a Binomial Random Variable According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a simple random sample of 400 car-owning households, determine the mean and standard deviation number of car-owning households that will have three or more cars. © 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Constructing Binomial Probability Histograms (a) Construct a binomial probability histogram with n = 8 and p = 0.15. (b) Construct a binomial probability histogram with n = 8 and p = 0. 5. (c) Construct a binomial probability histogram with n = 8 and p = 0.85. For each histogram, comment on the shape of the distribution. © 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment According to the Experian Automotive, 35% of all car-owning households have three or more cars. A researcher believes this percentage is higher than the percentage reported by Experian Automotive. He conducts a simple random sample of 400 car-owning households and found that 162 had three or more cars. Is this result unusual ? The result is unusual since 162 > 159.1 © 2010 Pearson Prentice Hall. All rights reserved