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1 Bluman, Chapter 5 1

2 guessing Suppose there is multiple choice quiz on a subject you don t know anything about. 15 th Century Russian Literature; Nuclear physics etc. You have to guess on every question. There are 5 questions and each question has 4 choices. Bluman, Chapter 5 2

3 Let x be the score on the test. Find p(x=0) In another words the probability you will get a score of zero, i.e. you will get all the questions wrong Find p(x=1) In another words the probability you will get a score of 1, i.e. you will get only one question correct. Bluman, Chapter 5 3

4 Question number Correct or wrong Bluman, Chapter 5 4

5 Repeat the process: P(2)= P(3)= p(4)= P(5)= Bluman, Chapter 5 5

6 What if the number of questions changed Let s say now the test has 10 questions and each question has 4 choices. What does the probability distribution chart looks like? Bluman, Chapter 5 6

7 x P(x) Bluman, Chapter 5 7

8 What if the number of choices changes Let s say now the test has 10 questions and each question has 5 choices. What does the probability distribution chart looks like? Bluman, Chapter 5 8

9 Bluman, Chapter 5 9

10 5-3 The Binomial Distribution Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes. Examples include: when a coin is tossed it can land on heads or tails, when a baby is born it is either a boy or girl. It will rain or it won t A person will pass the bar exam or not. 10

11 The Binomial Distribution The binomial experiment is a probability experiment that satisfies these requirements: 1. Each trial can have only two possible outcomes success or failure. 2. There must be a fixed number of trials. 3. The outcomes of each trial must be independent of each other. 4. The probability of success must remain the same for each trial. Bluman, Chapter 5 11

12 Notation for the Binomial Distribution P(S) The symbol for the probability of success P(F) The symbol for the probability of failure p The numerical probability of success q The numerical probability of failure P(S) = p and P(F) = 1 p = q n The number of trials X The number of successes Note that X = 0, 1, 2, 3,...,n Bluman, Chapter 5 12

13 The Binomial Distribution In a binomial experiment, the probability of exactly X successes in n trials is n! P X p q n - X! X! or number of possible desired outcomes X n X X P X C p q n x nx probability of a desired outcome Bluman, Chapter 5 13

14 Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-16 Page #272 Bluman, Chapter 5 14

15 Example 5-16: Survey on Doctor Visits A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. n! P X p q n - X! X! P ! 1 4 7!3! 5 5 X n X n 10,"one out of five" p, X Bluman, Chapter 5 15

16 Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-17 Page #273 Bluman, Chapter 5 16

17 Example 5-17: Survey on Employment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs. n 5, p 0.30,"at least 3" X 3, 4,5 5! P P X 2!3! 5! P !4! 5! P !5! Bluman, Chapter

18 Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-18 Page #273 Bluman, Chapter 5 18

19 Example 5-18: Tossing Coins A coin is tossed 3 times. Find the probability of getting exactly two heads, using Table B. n 3, p 0.5, X 2 P Bluman, Chapter 5 19

20 The Binomial Distribution The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas. Mean: np Variance: 2 npq Standard Deviation: npq Bluman, Chapter 5 20

21 Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-23 Page #276 Bluman, Chapter 5 21

22 Example 5-23: Likelihood of Twins The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins. np npq npq Bluman, Chapter 5 22

23 Tech notes Read technology notes on page 281. Read example 5-19 on page 274 Exercises 5.3 Page 276 #1, 5, 11, 15 and 17 Bluman, Chapter 5 23

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