Discrete Probability Distributions Chapter 6 Dr. Richard Jerz 1 GOALS Define the terms probability distribution and random variable. Distinguish between discrete and continuous probability distributions. Calculate the mean, variance, and standard deviation of a discrete probability distribution. Describe the characteristics of and compute probabilities using the binomial probability distribution. 2 What is a Probability Distribution? Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero heads, one head, two heads, and three heads. What is the probability distribution for the number of heads? 3 1
A Distribution Count 10 9 8 7 6 5 4 3 2 1 0 # of Green M&M's in Bag 9 3 3 2 1 0 1 2 3 4 5 6 # in Bag 4 A Probability Distribution Percent Probability of Green M&M's in Bag 60% 50.0% 50% 40% 30% 20% 16.7% 16.7% 11.1% 10% 5.6% 0.0% 0.0% 0% 1 2 3 4 5 6 7 # in Bag 5 Probability for Dice Toss 6 2
Tossing 2 Dice 7 Probability Distribution, # Heads, 3 Coin Tosses 8 Characteristics of a Probability Distribution The probability of a particular outcome is between 0 and 1 inclusive The outcomes are mutually exclusive events The list is exhaustive, the sum of the probabilities of the various events is equal to 1 9 3
Random Variables Random variable - a quantity resulting from an experiment that, by chance, can assume different values. 10 Types of Random Variables Discrete Random Variable can assume only certain clearly separated values. It is usually the result of counting something Continuous Random Variable can assume an infinite number of values within a given range. It is usually the result of some type of measurement 11 Discrete Random Variables Examples The number of students in a class. The number of children in a family. The number of cars entering a carwash in a hour. Number of home mortgages approved by Coastal Federal Bank last week. 12 4
Continuous Random Variables Examples The distance students travel to class. The time it takes an executive to drive to work. The length of an afternoon nap. The length of time of a particular phone call. 13 Characteristics of a Discrete Distribution The main features of a discrete probability distribution are: The sum of the probabilities of the various outcomes is 1.00. The probability of a particular outcome is between 0 and 1.00. The outcomes are mutually exclusive. and the item being measured can assume only certain separated (counted) values. 14 The Mean of a Probability Distribution The mean is a typical value used to represent the central location of a probability distribution. The mean of a probability distribution is also referred to as its expected value. 15 5
Similar to Weighted Mean The weighted mean of a set of numbers X 1, X 2,..., X n, with corresponding weights w 1, w 2,...,w n, is computed from the following formula: 16 The Variance and Standard Deviation Measures the amount of spread in a distribution The computational steps are: 1. Subtract the mean from each value, and square this difference. 2. Multiply each squared difference by its probability. 3. Sum the resulting products to arrive at the variance. The standard deviation is found by taking the square root of the variance. 17 Mean, Variance, and Standard Deviation Example John Ragsdale sells new cars for Pelican Ford. John usually sells the largest number of cars on Saturday. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday. 18 6
Mean of a Probability Distribution Example 19 Variance and Standard Deviation Example 20 Binomial Probability Distribution (discrete) Characteristics: There are only two possible outcomes on a particular trial of an experiment. The outcomes are mutually exclusive, The random variable is the result of counts. Each trial is independent of any other trial Examples: Yes or no True or false On or off Correct or incorrect 21 7
Tree Diagrams 22 Binomial Tree 23 Binomial Probability Formula Where C denotes a combination. n is the number of trials x is the random variable defined as the number of successes. π is the probability of a success on each trial 24 8
Example with M&M s Brown M&M (Binomial) 1.00 Probability 0.80 0.60 0.40 0.20 0.00 0 >0 2 Conditions 25 Question: What is the probability that 2 out of 4 bags have brown M&M s? N=4, x=2, π =.86, How about 3 out of 4 bags having brown M&M s? N=4, x=3, π =.86 How about between 2 and 3 out of 4 bags having brown M&M s? N=4, x=3?, π =.86 26 Example: Binomial Probability There are five flights daily from Pittsburgh via US Airways into the Bradford Pennsylvania Regional Airport. Suppose the probability that any flight arrives late is.20. What is the probability that none of the flights are late today? 27 9
Binomial Distribution Mean and Variance Mean of a binomial distribution Variance of a binomial distribution 28 Binomial Dist. Mean and Variance: Example For the example regarding the number of late flights, recall that π =.20 and n = 5. What is the average number of late flights? What is the variance of the number of late flights? 29 Binomial Dist. - Mean and Variance: Another Solution 30 10
Binomial Distribution - Table Five percent of the worm gears produced by an automatic, high-speed Carter-Bell milling machine are defective. What is the probability that out of six gears selected at random none will be defective? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five? Exactly six out of six? 31 Binomial Shapes for Varying π (n constant) 32 Binomial Shapes for Varying n with (π constant) 33 11
Cumulative Binomial Probability Distributions A study in June 2003 by the Illinois Department of Transportation concluded that 76.2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected. What is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts? 34 Binomial Probability - Excel 35 Cumulative Binomial Probability Dist. In Excel 36 12