Longwood University 201 High Street Farmville, VA 23901 Statistics Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

Longwood University 201 High Street Farmville, VA 23901 Discrete Probability Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

Discrete Probability Distributions CHARACTERISTICS OF A PROBABILITY DISTRIBUTION The probability of a particular outcome is between 0 and 1 inclusive. 2. The outcomes are mutually exclusive events. 3. The list is exhaustive. So the sum of the probabilities of the various events is equal to 1. DISCRETE RANDOM VARIABLE - A variable that can assume only certain clearly separated values and is the typically the result of counting something. CONTINUOUS RANDOM VARIABLE – A variable that can assume an infinite number of values within a given range. It is usually the result of some type of measurement

Probability Distributions PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome. 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? 6-4

Discrete Probability Distribution Mean X P(x) .05 .20 .06 .07 .08 .09

Discrete Probability Distribution Variance X P(x) .05 .20 .06 .07 .08 .09

Binomial Probability Distribution Discrete Probability Distribution Binomial Probability Distribution A Widely occurring discrete probability distribution Characteristics of a Binomial Probability Distribution 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 6-7

Binomial Distribution

Binomial Distribution Probability Distribution of Number of Heads Observed in 3 Tosses of a Coin 6-9

Binomial Distribution Calculating means and variances of a binominal distribution

Poisson Probability Distribution Poisson Distribution Poisson Probability Distribution The Poisson probability distribution is characterized by the number of times an event happens during some interval or continuum. Examples include: • The number of misspelled words per page in a newspaper. • The number of calls per hour received by Dyson Vacuum Cleaner Company. • The number of vehicles sold per day at Hyatt Buick GMC in Durham, North Carolina. • The number of goals scored in a college soccer game. 6-11

Poisson Distribution

Poisson Distribution Assume baggage is rarely lost by Northwest Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean number of lost bags per flight is 0.3 (300/1,000). If the number of lost bags per flight follows a Poisson distribution with u = 0.3, find the probability of not losing any bags. 6-13

Binomial – Shapes for Varying n ( constant) Binomial Distribution Binomial – Shapes for Varying n ( constant) 6-14

Poisson Distribution 6-15

Discrete Probability Distributions Problem: The arrival of customers at a service desk follows a Poisson distribution. If they arrive at a rate of two every five minutes, what is the probability that no customers arrive in a five-minute period?   .1353

Example Problem: Elly's hot dog emporium is famous for its chilidogs. Elly's latest sales indicate that 30% of the customers order their chilidogs with hot peppers. Suppose 18 customers are selected at random. What is the probability that exactly ten customers will ask for hot peppers?   0.015

Example Problem: For the following probability distribution, what is the variance?

Example Problem: There are eight flights from Minneapolis to St. Cloud each day. The probability that any one flight is late is 0.10. Using the binomial probability formula, what is the probability that 1 or more are late?  .43/.57

Example Problem: There are eight flights from Minneapolis to St. Cloud each day. The probability that any one flight is late is 0.10. Using the binomial probability formula, what is the probability that none are late?  .43/.57

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