1. Probability Distributions: Binomial & Normal Ginger Holmes Rowell, PhD MSP Workshop June 2006

2. Overview  Some Important Concepts/Definitions Associated with Probability Distributions  Discrete Distribution Example: Binomial Distribution More practice with counting and complex probabilities  Continuous Distribution Example: Normal Distribution

3. Start with an Example  Flip two fair coins twice List the sample space: S = { HH, HT, TH, TT } Define X to be the number of Tails showing in two flips. List the possible values of X = {0, 1, 2} Find the probabilities of each value of X

4. Use the Table as a Guide x Probability of getting “x” 0 1 2

5. Use the Table as a Guide x = # of tails Probability of getting “x” 0 P(X=0) = P(HH) =.25 1 P(X=1) = P(HT or TH) =.5 2 P(X=0) = P(HH) =.25

6. Draw a graph representing the distribution of X (# of tails in 2 flips)

7. Some Terms to Know  Random Experiment  Random Variable Discrete Random Variable Continuous Random Variable  Probability Distribution

8. Terms  Random Experiment: an experiment that can be repeated under the same conditions and you do not outcome of the experiment in advance  Examples: Flip a coin Roll a pair of dice Survival times of persons with a given disease

9. Terms Continued  Random Variable: a numerical representation of the outcome of a random experiment  Examples Flip a coin twice: X counts the number of tails showing Rolling two dice: X represents the sum of the two faces showing Survival times: number of months people live once diagnosed

10. Terms Continued  Discrete Random Variable A RV that has a finite (or countable) number of possible outcomes  Example Sum of faces showing when roll two dice  Continuous Random Variable A RV that has an infinite (uncountable) number of possible outcomes  Example Birth weight, time spent doing homework

11. Terms Continued  The Probability Distribution of a random variable, X, is a table, chart, graph or formula which specifies the probabilities for all possible values in the sample space (i.e. for all possible values of the random variable).  Example: Let’s go back to our original example of flipping a fair coin twice.

12. X counts the number of tails in two flips of a coin x Probability of getting “x” 0 P(X=0) =.25 1 P(X=1) =.50 2 P(X=2) =.25 Specify the random experiment & the random variable for this probability distribution. Is the RV discrete or continuous?

13. Properties of Discrete Probability Distributions  The sum of the probabilities of all items equals one  Each individual item’s probability is between 0 and 1 (inclusive)  Example: Binomial Distribution  Probability Histogram Horizontal axis shows values of the RV & vertical axis represents the probability that the corresponding value of the RV occurs.

14. Mean of a Discrete RV  Mean value = sum of ( value of x * probability the given value occurs )  Example: X counts the number of tails showing in two flips of a fair coin Mean = sum of [ x*P(X=x) ] over all x’s = 0*.25 + 1*.5 + 2 *.25 = 1 tail showing

15. Example: Your Turn  Example # 12, parental involvement

16. Overview  Some Important Concepts/Definitions Associated with Probability Distributions  Discrete Distribution Example: Binomial Distribution More practice with counting and complex probabilities  Continuous Distribution Example: Normal Distribution

17. Binomial Distribution  If X counts the number of successes in a binomial experiment, then X is said to be a binomial RV. A binomial experiment is a random experiment that satisfies the following Each trial ends in a success or failure P(Success) is the same for every trial There are a finite number of trials Trials are independent.

18. Binomial Example  Flip a fair coin twice  Rolling doubles on two dice  Shooting 10 free throws  …

19. What is the Binomial Probability Distribution?  Example: Flip a fair coin twice We already answered this problem  Example: Test Guessing Handout  Generate General Formula

20. Binomial Distribution  Let X count the number of successes in a binomial experiment which has n trials and the probability of success on any one trial is represented by p, then Check for the last example: P(X = 2) = ____

21. Mean of a Binomial RV  Example: Test guessing  In general: mean = n*p  Variance = n*p*(1-p)

22. Using the TI-84  To find P(X=a) for a binomial RV for an experiment with n trials and probability of success p  Binompdf(n, p, a) = P(X=a)  Binomcdf(n, p, a) = P(X <= a)

23. Pascal’s Triangle & Binomial Coefficients  Handout  Pascal’s Triangle Applet http://www.mathforum.org/dr.cgi/pascal.cgi ?rows=10 http://www.mathforum.org/dr.cgi/pascal.cgi ?rows=10

24. Using Tree Diagrams for finding Probabilities of Complex Events  For a one-clip paper airplane, which was flight-tested with the chance of throwing a dud (flies < 21 feet) being equal to 45%. What is the probability that exactly one of the next two throws will be a dud and the other will be a success?

25. Airplane Example Source: NCTM Standards for Prob/Stat. D:\Standards\document\chapter6\data.htm

26. Airplane Problem  A: Probability = 198/400, or.495, since each of the two possibilities—"dud first, then success" and "success first, then dud"—has a probability of 99/400.

27. Homework  Blood type problem  Handout # 22, 26, 37

28. Overview  Some Important Concepts/Definitions Associated with Probability Distributions  Discrete Distribution Example: Binomial Distribution More practice with counting and complex probabilities  Continuous Distribution Example: Normal Distribution

29. Continuous Distributions  Probability Density Function An equation used to compute probabilities of continuous random variables that satisfy the following  Area under the graph of the equation over all the possible values of the RV must equal one  The graph of the equation must be >= zero for all possible values of the RV

30. Example: Normal Distribution  Draw a picture  Show Probabilities  Show Empirical Rule

31. What is Represented by a Normal Distribution?  Yes or No Birth weight of babies born at 36 weeks Time spent waiting in line for a roller coster on Sat afternoon? Length of phone calls for a give person IQ scores for 7 th graders SAT scores of college freshman

32. Penny Ages  Collect pennies with those at your table.  Draw a histogram of the penny ages  Describe the basic shape  Do the data that you collected follow the empirical rule?

33. Penny Ages Continued  Based on your data, what is the probability that a randomly selected penny is is between 5 & 10 years old? Is at least 5 years old? Is at most 5 years old? Is exactly 5 years old? Find average penny age & standard deviation of penny age

34. Using your calculator  Normalcdf ( a, b, mean, st dev)  Use the calculator to solve problems on the previous page.

35. Homework  Handout #’s 12, 14, 15, 16, 24