1. Chapter 5 Discrete Probability Distributions

2. Introduction Many decisions in real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation, then evaluating the results Decisions can be made using statistics such as mean, variance, and standard deviation

3. 5.1 – Probability Distributions Random Variable Variable whose values are determined by chance Two types of random variables 1.Discrete variables Finite number of possible values or an infinite number of values that can be counted 2.Continuous variables Obtained from data that can be measured rather than counted

4. Probability Distribution Discrete probability distribution Consists of values a random variable can assume and corresponding probabilities of values Probabilities are determined theoretically or by observation Examples 5-1 Construct a probability distribution for rolling a single die 5-2 Represent graphically probability distribution for sample space for tossing three coins

5. Examples 5-3 The baseball World Series is played by the winner of the National League and the American League. The first team to win four games wins the World Series. In other words, the series will consist of four to seven games, depending on the individual victories. The data shown consist of the number of games played in the WS from 1965 to 2005. The number of games played is represented by the variable X. Find the probability P(X) for each X, construct a probability distribution, and draw a graph for the data

6. Example 5-3 XNumber of games played 48 57 69 716

7. Requirements for a PD

8. Example 5-4 Determine whether each distribution is a probability distribution X05101520 P(X)1/5 X0246 P(X)1.50.30.2 X1234 P(X)¼1/81/169/16 X237 P(X)0.50.30.4

9. 5.2 – Mean, Variance, Standard Deviation, and Expectation Mean, variance, and standard deviation for a probability distribution are computed differently from mean, variance, and standard deviation for samples

10. Mean

11. Examples 5-5 Find the mean of the number of spots that appear when a standard die (6-sided) is tossed 5-6 In a family with two children, find the mean of the number of children who will be girls 5-7 If three coins are tossed, find the mean of the number of heads that occur

12. Example 5-8 The probability distribution shown represents the number of trips of five nights or more that American adults take per year. Find the mean Number of heads X Probability P(X) 00.06 10.70 20.20 30.03 40.01

13. Variance & Standard Deviation

14. Examples 5-9 Compute the variance and standard deviation for the probability distribution in example 5-5 5-10 A box contains 5 balls. Two are numbered 3, one is numbered 4, and two are numbered 5. the balls are mixed and one is selected at random. After a ball is selected, its number is recorded. Then it is replaced. If the experiment is repeated many times, find the variance and standard deviation of the numbers on the balls.

15. Expectation

16. Example 5-12 One thousand tickets are sold at $1 each for a color TV valued at $350. What is the expected value of the gain if you purchase one ticket?

17. Example 5-14 A financial adviser suggests that his client select one of two types of bonds in which to invest $5000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a 2.5% return and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. When the bond defaults, the investor loses all the investment.

18. 5.3 – Binomial Distribution Many probability problems have two outcomes or can be reduced to two outcomes

19. Binomial Experiments Binomial experiment Probability experiment that satisfies the following four requirements: 1.There must be a fixed number of trials 2.Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure 3.The outcomes of each trial must be independent of one another 4.The probability of a success must remain the same for each trial

20. Binomial Distribution Binomial distribution Outcomes of a binomial experiment and the corresponding probabilities of these outcomes

21. Notation for Bin. Distribution

22. Bin. Probability Formula

23. Examples 5 – 15 A coin is tossed 3 times. Find the probability of getting exactly two heads 5 – 16 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

24. Example 5 - 17 A survey from Teenage Research Unlimited 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

25. Statistics for Bin. Distribution

26. Examples 5 – 21 A coin is tossed 4 times. Find the mean, variance, and standard deviation of the number of heads that will be obtained 5 – 22 A dis is rolled 360 times. Find the mean, variance, and standard deviation of the number of 4s that will be rolled

27. 5.4 – Poisson Distribution Poisson distribution A discrete probability distribution that is useful when n is large and p is small and when the independent variables occur over a period of time Can be used when a density of items is distributed over a given area or volume, such as the number of plants per acre or the number of defects in a given length of videotape

28. Formula for Poisson Dist.

29. Examples 5 – 27 If there are 200 typographical errors randomly distributed in a 500-page manuscript, find the probability that a given page contains exactly 3 errors 5 – 28 A sales firm receives, on average, 3 calls per hour on its toll-free number. For any given hour, find the probability that it will receive the following: a)At most 3 calls b)At least 3 calls c)5 or more calls

30. Example 5 – 29 If approximately 2% of the people in a room of 200 people are left-handed, find the probability that exactly 5 people there are left-handed