Binomial Distributions

Section 4.2

• There are a fixed number of trials. (n)• The n trials are independent and repeated under identical

conditions.• Each trial has 2 outcomes, S = Success or F = Failure.• The probability of success on a single trial is p. P(S) = p The probability of failure is q. P(F) =q where p + q = 1• The central problem is to find the probability of x successes out

of n trials. Where x = 0 or 1 or 2 … n.

Binomial ExperimentsCharacteristics of a Binomial Experiment

The random variable x is a count of the number of successes in n trials.

1. What is the 11th digit after the decimal point for the irrational number e?(a) 2 (b) 7 (c) 4 (d) 5

2. What was the Dow Jones Average on February 27, 1993?(a) 3265 (b) 3174 (c) 3285 (d) 3327

3. How many students from Sri Lanka studied at U.S. universities from 1990-91?(a) 2320 (b) 2350 (c) 2360 (d) 2240

4. How many kidney transplants were performed in 1991?(a) 2946 (b) 8972 (c) 9943 (d) 7341

5. How many words are in the American Heritage Dictionary?(a) 60,000 (b) 80,000 (c) 75,000 (d) 83,000

Guess the Answers

Quiz Results

Count the number of correct answers. Let the number of correct answers = x.

Why is this a binomial experiment?

What are the values of n, p and q?

What are the possible values for x?

The correct answers to the quiz are:1. d 2. a 3. b 4. c 5. b

A multiple choice test has 8 questions each of which has 3 choices, one of which is correct. You want to know the probability that you guess exactly 5 questions correctly.Find n, p, q, and x.

A doctor tells you that 80% of the time a certain type of surgery is successful. If this surgery is performed 7 times, find the probability exactly 6 surgeries will be successful. Find n, p, q, and x.

n = 8 p = 1/3 q = 2/3 x = 5

n = 7 p = 0.80 q = 0.20 x = 6

Binomial Experiments

Try It YourselfDecide whether the following is a Binomial experiment. If it is specify the values of n, p, and q And list the possible values of the random variable x. If it is not explain why. You take a multiple choice quiz that consists of 10 questions. Each question has 4 possible answers only one of which is correct. To complete the quiz, you randomly guess the answer to each question. The random variable represents the number of correct answers.a. Identify a trial of the experiment and what is a success.b. Decide if the experiment satisfies the four conditions of a

binomial experiment.c. Make a conclusion and identify n,p, q and the possible

values of x.

5. Cyanosis is the condition of having blush skin due to insufficient oxygen to the blood. A bout 80% of the babies born with Cyanosis recover fully. A hospital is caring for 5 babies born with Cyanosis. The random variable represents the number of babies born that recover fully.Is this a binomial experiment?If so what is n, p, q and what are the random values of x?

Find the probability of getting exactly 3 questions correct on the quiz.

Write the first 3 correct and the last 2 wrong as SSSFFP(SSSFF) = (.25)(.25)(.25)(.75)(.75) = (.25)3(.75)2 = 0.00879

Since order does not matter, you could get any combination of three correct out of five questions. List these combinations.

SSSFF SSFSF SSFFS SFFSS SFSFSFFSSS FSFSS FSSFS SFSSF FFSSF

Each of these 10 ways has a probability of 0.00879.

P(x = 3) = 10(0.25)3(0.75)2 = 10(0.00879) = 0.0879

Binomial Probabilities

Binomial Probability FormulaIn a Binomial experiment the probability of exactly x successes in n trials is P(X) = n C x px qn-x = n! = px qn-x (n-x)!x!

Find the probability of getting exactly 3 questions correct on the quiz.

Each of these 10 ways has a probability of 0.00879.

P(x = 3) = 10(0.25)3(0.75)2= 10(0.00879)= 0.0879

Combination of n values, choosing x

There are ways.

Binomial ProbabilitiesIn a binomial experiment, the probability of exactly x successes in n trials is

Use the formula to calculate the probability of getting none correct, exactly one, two, three, four correct or all 5 correct on the quiz.

P(3) = 0.088 P(4) = 0.015 P(5) = 0.001

Binomial Distribution

0 1 2 3 4 5

0

.10

.20

.30

.40

.237

.396

.294

.088

.015 .001

x P(x)0 0.2371 0.3962 0.2643 0.0884 0.0155 0.001

Binomial Histogram

x

Probabilities

1. What is the probability of answering either 2 or 4 questions correctly?

2. What is the probability of answering at least 3 questions correctly?

3. What is the probability of answering at least one question correctly?

P( x = 2 or x = 4) = 0.264 + 0.015 = 0. 279

P(x 3) = P( x = 3 or x = 4 or x = 5) = 0.088 + 0.015 + 0.001 = 0.104

P(x 1) = 1 - P(x = 0) = 1 - 0.237 = 0.763

x P(x)0 0.2371 0.3962 0.2643 0.0884 0.0155 0.001

Parameters for a Binomial Experiment

Use the binomial formulas to find the mean, variance and standard deviation for the distribution of correct answers on the quiz.

Mean:

Variance:

Standard deviation:

9.) Find the mean, Variance, and Standard deviation of the Binomial distribution when n =100, p = 0.4Mean = = 100 (0.4) = 40Variance = q= 1- p = 1- 0.4 = .6 100(.4)(.6) = 24Stand. Deviation = = =4.89

24

Home wk: 1-12 all Pg. 193-194 Day 2 Home wk: 13-26 all Pg. 194-196

Answers to 1.) (a) p= 0.50 (b) p = 0.20, (c ) p=0.802.) (a) p=0.75, (b) p = 0.50, ( c ) p= 0.253.) (a) n= 12 , (b) n = 4, ( c ) n= 8As n increases the distribution becomes more symmetric.4.) (a) n= 10 , (b) n = 15, ( c ) n= 5As n increases the distribution becomes more symmetricSee #5 Above6.) a.) Yes Binomial Experiment n =18, p = 74%, q = 26% , x = 0,1,2…187.) No this is not a binomial Experiment there are more than 2 possible outcomes for each trial.8.) No this is not a binomial Experiment because the probability for success is not the same for each trial.9.) 40;24;4.9 (10) 60.8; 12.16; 3.48711.) 22.08; 18.547; 4.307 (12.) 215.46; 79.720; 8.929