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# Probability 4.2

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### Probability 4.2

1. 1. Binomial Distributions Chapter 4.2
2. 2. Objectives <ul><li>Learn to determine if a probability experiment is a binomial experiment </li></ul><ul><li>Learn to find binomial probabilities using the binomial probability formula </li></ul><ul><li>Learn to find binomial probabilities using technology, formulas, and a binomial probability table </li></ul><ul><li>Learn to graph a binomial distribution </li></ul><ul><li>Learn to find the mean, variance, & standard deviation of a binomial probability distribution. </li></ul>
3. 3. What does “binomial” mean?
4. 4. Binomial Experiments <ul><li>A binomial experiment is a probability experiment that: </li></ul><ul><ul><li>Is repeated for a fixed number of trials, where each trial is independent of the others </li></ul></ul><ul><ul><li>Has only 2 possible outcomes of interest for each trial. ( Success or Failure ) </li></ul></ul><ul><ul><li>Has the same probability of success for each trial: P(S) = P(F) </li></ul></ul><ul><ul><li>Has a random variable x that counts the number of successful trials. </li></ul></ul>
5. 5. Binomial Experiment Notation Symbol Description n The number of times a trial is repeated p = P(S) The probability of success in a single trial q = 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
6. 6. Drawing a Club Example <ul><li>From a standard deck of cards, pick a card, note whether it is a club or not, and replace the card. </li></ul><ul><li>Repeat the experiment 5 times. (n = 5) </li></ul><ul><li>S = selecting a club, F = not selecting a club </li></ul><ul><li>p = P(S) = </li></ul><ul><li>¼ </li></ul><ul><li>q = P(F) = </li></ul><ul><li>¾ </li></ul><ul><li>x = number of clubs out of 5 </li></ul><ul><li>Possible values of x are: </li></ul><ul><li>0, 1, 2, 3, 4, and 5 </li></ul>
7. 7. Binomial or not? <ul><li>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. </li></ul><ul><li>Yes </li></ul>
8. 8. Binomial or not? <ul><li>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. </li></ul><ul><li>No </li></ul>
9. 9. Binomial or not? <ul><li>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. </li></ul><ul><li>What is a trial of this experiment? </li></ul><ul><li>What is a “success”? </li></ul><ul><li>Does this experiment satisfy the 4 conditions of a binomial experiment? </li></ul><ul><li>What are n, p, q, and the possible values of x? </li></ul>
10. 10. What is the probability? <ul><li>If you roll a 6-sided die 3 times, what is the probability of getting exactly one 6? </li></ul>
11. 11. Binomial Probability Formula <ul><li>In a binomial experiment, the probability of exactly x successes in n trials is: </li></ul><ul><li>P(x) = n C x p x q n-x = _ n! _ p x q n-x </li></ul><ul><li>(n-x)!x! </li></ul>
12. 12. What is the probability? <ul><li>If you roll a 6-sided die 3 times, what is the probability of getting exactly one 6? </li></ul><ul><li>P(x) = n C x p x q n-x = _ n! _ p x q n-x </li></ul><ul><li>(n-x)!x! </li></ul><ul><li>N = 3, p = 1/6, q = 5/6, x = 1 </li></ul><ul><li>P(1) = 3! (1/6) 1 *(5/6) 2 </li></ul><ul><li>(3-1)!1! </li></ul><ul><li>= 3 * (1/6) * (25/36) = 25/72 = .347 </li></ul>
13. 13. Another example . . . <ul><li>54% of men consider themselves to be basketball fans. You randomly select 15 men and ask each if he considers himself a basketball fan. </li></ul><ul><li>Find the probability that the number who consider themselves basketball fans is </li></ul><ul><ul><li>Exactly 8 </li></ul></ul><ul><ul><li>At least 8 </li></ul></ul><ul><ul><li>Less than 8 </li></ul></ul>
14. 14. Technology to the Rescue! <ul><li>To use Excel to find the answer, use: </li></ul><ul><li>=Binomdist(x,n,p,TRUE/FALSE) </li></ul><ul><li>Where FALSE means “exactly x” and TRUE means “no more than x” </li></ul><ul><li>Binomial Basketball.xlsx </li></ul>
15. 15. Using Binomial Probability Tables <ul><li>Turn to the back of your books, p. A8, Table 2 in Appendix B </li></ul><ul><li>Find n, x, and p </li></ul><ul><li>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? </li></ul><ul><li>What are n, p, and x? </li></ul><ul><li>10, .30, 4 </li></ul><ul><li>What is the answer? </li></ul><ul><li>.2 </li></ul>
16. 16. Graphing Binomial Distributions <ul><li>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. </li></ul><ul><li>Using n = 6, p = .65, and q = .35, you get: </li></ul>x 0 1 2 3 4 5 6 P(x) .002 .020 .095 .235 .328 .244 .075
17. 17. Graphing Binomial Distributions x 0 1 2 3 4 5 6 P(x) .002 .020 .095 .235 .328 .244 .075
18. 18. Mean, Variance, & Standard Deviation <ul><li>You can use the formulas from section 4.1, but these are easier: </li></ul><ul><li>Mean: μ = np </li></ul><ul><li>Variance: σ 2 = npq </li></ul><ul><li>Standard Deviation: σ = √npq </li></ul>
19. 19. Homework <ul><li>P. 193 Do 1-9 together </li></ul><ul><li>P. 194 10-24 evens </li></ul>