The Binomial Probability Distribution Section 6.2

Objectives Determine whether a probability experiment is a binomial experiment Compute probabilities of binomial experiments Compute the mean and standard deviation of a binomial random variable Construct binomial probability histograms

Characteristics of Binomial Probability Experiments Binomial Probability Distribution: Discrete Describes probabilities for experiments in which there are only 2 mutually exclusive (disjoint) outcomes Examples: yes/no, heads/tails,

Criteria for Binomial Probability Experiments An experiment is said to be a binomial experiment if: Performed a fixed number of times Each time is called a trial The trials are independent The outcome of one trial will not affect the probability of the outcome of any of the other trials For each trial, there are only 2 mutually exclusive (disjoint) outcomes: success or failure The probability of success is the same for each trial of the experiment

Notation Used with the Binomial Probability Distribution As always, a capital letter, X, is the notation for a random variable There are always n independent trials of the experiment Let p denote the probability of success, so that 1 – p is the probability of failure Let x denote the number of successes in n independent trials of the experiment. So 0 < x < n

Examples Examples of binomial experiments Tossing a coin 30 times to see how many heads occur. Asking 100 people if they watch CNN news. Rolling a die to see if a 2 appears. Examples which aren't binomial experiments Rolling a die until a 4 appears (not a fixed number of trials) Asking 50 people how old they are (not two outcomes) Drawing 5 cards from a deck for a poker hand (done without replacement, so not independent) http://people.richland.edu/james/lecture/m170/ch06-bin.html

Compute Probabilities of Binomial Experiments

Compute Probabilities of Binomial Experiments

Binomial Probability Distribution Function

Using the Binomial Probability Distribution Function A coin is tossed 10 times. What is the probability that exactly 6 heads will occur?

Using the Binomial Probability Distribution Function A coin is tossed 10 times. What is the probability that at least 5 heads will occur?

Using the Binomial Probability Distribution Function A coin is tossed 10 times. What is the probability that at fewer than 3 heads will occur?

Using the Binomial Probability Distribution Function A coin is tossed 10 times. What is the probability that 3 - 5 heads will occur?

Assignment Pg 340: 1, 3, 4, 5, 6, 7-16, 17, 20, 23, 25, 28

Computing Probabilities Using the Binomial Table A coin is tossed 10 times. What is the probability that exactly 6 heads will occur? http://images.tutorvista.com/cms/images/67/binomial-distribution-table.jpg

Computing Using the TI-83/84 http://cfcc.edu/faculty/cmoore/TIBinomial.htm

Computing Using the TI-83/84 http://cfcc.edu/faculty/cmoore/TIBinomial.htm

Computing Using the TI-83/84 http://cfcc.edu/faculty/cmoore/TIBinomial.htm

Compute the Mean and Standard Deviation of a Binomial Random Variable These formulas are very simple and therefore there are no commands on the TI-83/84 calculator to replicate this.

Example 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.

Construct Binomial Probability Histograms Construct a binomial probability histogram with n=10 and p=0.2. What is the shape of this histogram? x P(x) 1 2 3 4 5 6 7 8 9 10

Examples (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.

n = 8 and p = 0.15

n = 8 and p = 0. 5

n = 8 and p = 0.85

n = 50 and p = 0.8

n = 70 and p = 0.8

Number of Trials Affects Shape 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. http://phet.colorado.edu/en/simulation/plinko-probability

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

Assignment Pg 340-343: 29, 30, 34, 36, 42, 43, 45, 47, 54