#### Transcript Probability Distributions: Binomial & Normal

Probability Distributions: Binomial & Normal Ginger Holmes Rowell, PhD MSP Workshop June 2006 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 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 Use the Table as a Guide x 0 1 2 Probability of getting “x” Use the Table as a Guide x = # of Probability of getting “x” tails 0 P(X=0) = P(HH) = .25 1 P(X=1) = P(HT or TH) = .5 2 P(X=0) = P(HH) = .25 Draw a graph representing the distribution of X (# of tails in 2 flips) Some Terms to Know Random Experiment Random Variable Discrete Random Variable Continuous Random Variable Probability Distribution 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 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 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 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. X counts the number of tails in two flips of a coin x 0 Probability of getting “x” 1 P(X=1) = .50 2 P(X=2) = .25 P(X=0) = .25 Specify the random experiment & the random variable for this probability distribution. Is the RV discrete or continuous? 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. 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 Example: Your Turn Example # 12, parental involvement 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 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. Binomial Example Flip a fair coin twice Rolling doubles on two dice Shooting 10 free throws … What is the Binomial Probability Distribution? Example: Flip a fair coin twice We already answered this problem Example: Test Guessing Handout Generate General Formula 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 n a na P( X a) p (1 p) for x 0, 1, 2,...n a Check for the last example: P(X = 2) = ____ Mean of a Binomial RV Example: Test guessing In general: mean = n*p Variance = n*p*(1-p) 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) Pascal’s Triangle & Binomial Coefficients Handout Pascal’s Triangle Applet http://www.mathforum.org/dr.cgi/pascal.cgi ?rows=10 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? Airplane Example Source: NCTM Standards for Prob/Stat. D:\Standards\document\chapter6\data.htm 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. Homework Blood type problem Handout # 22, 26, 37 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 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 Example: Normal Distribution Draw a picture Show Probabilities Show Empirical Rule 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 7th graders SAT scores of college freshman 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? 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 Using your calculator Normalcdf ( a, b, mean, st dev) Use the calculator to solve problems on the previous page. Homework Handout #’s 12, 14, 15, 16, 24