Section 6.2 The Binomial Probability Distribution 6-1 1

Recall: A Discrete random variable can take whole (countable) values A Continuous random variable can take on any value between two specified values Examples: Flipping a coin a number of times: the Number of heads observed is a discrete variable. A fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds. This chapter deals exclusively with discrete random variables - experiments where the data observed is a ‘countable’ value. Give examples. Following chapters will deal with continuous random variables. Page 201 of Elementary Statistics, 10th Edition

Consequently: Two types of Probability Distributions Discrete Probability Distribution Continuous Probability Distribution This chapter deals exclusively with discrete random variables - experiments where the data observed is a ‘countable’ value. Following chapters will deal with continuous random variables. Page 201 of Elementary Statistics, 10th Edition

A Discrete Probability Distribution: The Binomial Distribution

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. 6-5 5

Notation Used in the Binomial Probability Distribution There are n independent trials of the experiment. p denotes the probability of success so 1 – p is the probability of failure. X is the binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 < X < n. 6

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. 6-7 7

Computing a Binomial Probability Value 6-8 8

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? 9

Computing the Mean and Standard Deviation of a Binomial Probability Distribution 6-10 10

EXAMPLE. Finding the Mean and Standard Deviation of a Binomial 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. 6-11 11

Constructing Binomial Probability Histograms EXAMPLES 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. 6-12 12

n=8, p=0.15 x P(x) 0.272491 1 0.384693 2 0.237604 3 0.08386 4 0.018499 5 0.002612 6 0.00023 7 1.16E-05 8 2.56E-07 13

n=8, p=0.5 x P(x) 0.003906 1 0.03125 2 0.109375 3 0.21875 4 0.273438 5 6 7 8 6-14 14

n=8, p=0.85 x P(x) 2.56289E-07 1 1.16184E-05 2 0.000230432 3 0.002611567 4 0.018498596 5 0.083860304 6 0.237604195 7 0.384692506 8 0.272490525 6-15 15

For a Larger number of trials, n=25: 6-16 16

n=50: 17

n=70: Demo 18

For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable X becomes bell-shaped. As a general rule of thumb, if np(1 – p) > 10, then the probability distribution will be approximately bell-shaped. 6-19 19

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EXAMPLE. Using the Mean, Standard Deviation and 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 (more than 2 standard dev. greater than the mean) 21