Binomial Distributions

Chapter 4.2

Objectives• Learn to determine if a probability experiment is a

binomial experiment• Learn to find binomial probabilities using the

binomial probability formula• Learn to find binomial probabilities using

technology, formulas, and a binomial probability table

• Learn to graph a binomial distribution• Learn to find the mean, variance, & standard

deviation of a binomial probability distribution.

What does “binomial” mean?

Binomial Experiments

• A binomial experiment is a probability experiment that:– Is repeated for a fixed number of trials, where each trial is

independent of the others– Has only 2 possible outcomes of interest for each trial.

(Success or Failure)– Has the same probability of success for each trial:

P(S) = P(F)– Has a random variable x that counts the number of

successful trials.

Binomial Experiment Notation

Symbol Descriptionn The number of times a trial is repeatedp = P(S) The probability of success in a single trialq = P(F) The probability of failure in a single trial (q = 1 – p)x The random variable represents a count of the

number of successes in n trials: x = 0, 1, 2, 3, . . ., n

Drawing a Club Example• From a standard deck of cards, pick a card, note

whether it is a club or not, and replace the card.• Repeat the experiment 5 times. (n = 5)• S = selecting a club, F = not selecting a club• p = P(S) = • ¼• q = P(F) =• ¾• x = number of clubs out of 5• Possible values of x are:• 0, 1, 2, 3, 4, and 5

Binomial or not?

• A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on 8 patients. The random variable represents the number of successful surgeries.

• Yes

Binomial or not?

• A jar contains 5 red marbles, 9 blue marbles, and 6 green marbles. You randomly select 3 marbles from the jar, without replacement. The random variable represents the number of red marbles.

• No

Binomial or not?

• You take a multiple choice quiz that consists of 10 questions. Each question has 4 possible answers, only 1 of which is correct. You randomly guess the answer to each question. The random variable represents the number of correct answers.

• What is a trial of this experiment?• What is a “success”?• Does this experiment satisfy the 4 conditions of a

binomial experiment?• What are n, p, q, and the possible values of x?

What is the probability?

• If you roll a 6-sided die 3 times, what is the probability of getting exactly one 6?

Binomial Probability Formula

• In a binomial experiment, the probability of exactly x successes in n trials is:

• P(x) = nCx pxqn-x =_ n! _ pxqn-x

• (n-x)!x!

What is the probability?

• If you roll a 6-sided die 3 times, what is the probability of getting exactly one 6?

• P(x) = nCx pxqn-x =_ n! _ pxqn-x

• (n-x)!x!• N = 3, p = 1/6, q = 5/6, x = 1• P(1) = 3! (1/6)1*(5/6)2

• (3-1)!1!• = 3 * (1/6) * (25/36) = 25/72 = .347

Another example . . .

• 54% of men consider themselves to be basketball fans. You randomly select 15 men and ask each if he considers himself a basketball fan.

• Find the probability that the number who consider themselves basketball fans is– Exactly 8– At least 8– Less than 8

Technology to the Rescue!

• To use Excel to find the answer, use:• =Binomdist(x,n,p,TRUE/FALSE)• Where FALSE means “exactly x” and TRUE

means “no more than x”• Binomial Basketball.xlsx

Using Binomial Probability Tables

• Turn to the back of your books, p. A8, Table 2 in Appendix B

• Find n, x, and p• 30% of all small businesses in the U.S. have a

website. If you randomly select 10 small businesses, what is the probability that exactly 4 of them have a website?

• What are n, p, and x?• 10, .30, 4• What is the answer?• .2

Graphing Binomial Distributions

• 65% of households in the U.S. subscribe to cable TV. You randomly select 6 households and ask if they subscribe to cable. Construct a probability distribution for the random variable x, and graph it.

• Using n = 6, p = .65, and q = .35, you get:

x 0 1 2 3 4 5 6

P(x) .002 .020 .095 .235 .328 .244 .075

Graphing Binomial Distributionsx 0 1 2 3 4 5 6

P(x) .002 .020 .095 .235 .328 .244 .075

Mean, Variance, & Standard Deviation

• You can use the formulas from section 4.1, but these are easier:

• Mean: μ = np• Variance: σ2 = npq• Standard Deviation: σ = √npq

Homework

• P. 193 Do 1-9 together• P. 194 10-24 evens