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Chapter 6 Discrete Probability Distributions

Chapter 6 Discrete Probability Distributions. North Seattle Community College BUS210 Business Statistics. Learning Objectives. In this section, you will learn about:. Probability distributions and their properties Important discrete probability distributions: The Binomial Distribution

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Chapter 6 Discrete Probability Distributions

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  1. Chapter 6 Discrete Probability Distributions North Seattle Community College BUS210 Business Statistics

  2. Learning Objectives In this section, you will learn about: • Probability distributions and their properties • Important discrete probability distributions: • The Binomial Distribution • The Poisson Distribution • Calculating the expected value and variance • Using the Binomial and Poisson distributions to solve business problems

  3. Definitions • Random variable: representation of a possible numerical value from an uncertain event. • Discreterandom variables: “How many” produce outcomes that come from counting (e.g. number of courses you are taking). • Continuousrandom variables: “How much” produce outcomes that come from a measurement (e.g. your annual salary, or your weight).

  4. Random Variables Random Variables Discrete Random Variable Discrete Random Variable Continuous Random Variable Ch. 7 Ch. 6

  5. Discrete Random Variables • Can only assume a countable number of values Examples: • Rolling five dice (as in Yahtzee) If X is the number of times that a 6 shows up, then X could be equal to 0, 1, 2, 3, 4, or 5 • Flipping a coin 3 times. If X is the number of times that heads comes up, then X could be equal to 0, 1, 2, or 3

  6. Discrete Random Variables • A probability distribution is… a mutually exclusive listing of all possible numerical outcomes for a variable and the probability of an occurrence associated with each outcome. • Example for a discrete random variable: Notice that the total probability adds up to 1.00

  7. Discrete Random Variable Probability Distribution Probability Distribution Experiment: Toss 2 Coins. Let X = # heads. T T 4 possible outcomes T H H T H H 0.50 0.25 Probability Total = 1.00 0 1 2 X

  8. Discrete Random Variable Expected Value Expected Value is the mean of a discrete random variable Also known as the Weighted Average • Example:Experiment of 2 Tossed Coins

  9. Discrete Random Variable Measuring Dispersion • Variance of a discrete random variable • Standard Deviation of a discrete random variable

  10. Discrete Random Variable Measuring Dispersion (continued) • Example: standard deviation of 2 tossed coins

  11. Probability Distributions Probability Distributions Discrete Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Binomial Binomial Binomial Normal Exponential Poisson Poisson Ch. 6 Ch. 7

  12. Probability Distributions Binomial • A finite number of observations, n • 20 tosses of a coin • 12 auto batteries purchased from a wholesaler • Each observation is categorized as… • Success: “event of interest”has occurred, or • i.e. - heads (or tails) in each toss of a coin; • i.e. - defective (or not defective) battery • Failure: “event of interest”has not occurred • The complement of a success • These two categories, success & failure, are • mutually exclusive and collectively exhaustive Note: A random experiment with only 2 outcomes is known as a Bernoulli experiment.

  13. Probability Distributions Binomial (continued) • The probability of… • success (event occurring) is represented as π • failure (event not occurring) is 1 – π • Probability of success (π) is constant • Probability of getting tails is the same for each toss of the coin • Observations are independent • Each event is unaffected by any other event • Two sampling methods deliver independence • Infinite population without replacement • Finite population with replacement

  14. Probability Distributions Binomial (continued) • Applications: Situations where are there only two outcomes • Apple marks newly manufactured iPads as either defective or acceptable • Visitors to Amazon’s website will either buy an item or not buy an item. • Asking if voters will approve a referendum, a pollster receives responses of “yes” or “no” • Federal Express marks a delivery as either damaged or not damaged

  15. Probability Distributions Binomial (continued) Counting Revisited: • You toss a coin three times. In how many ways can you get two heads? • Possible ways: HHT, HTH, THH, • so there are three ways you can getting two heads. • This situation is fairly simple….but, what about 10 times?…a 100?...or even a thousand? • For more complicated situations we need a method • to be able to count the number of ways (combinations).

  16. Counting TechniquesRule of Combinations • The number of combinations of selecting x objects out of n objects is where:n! means “n factorial” 2! = (1)(2) 4! = (1)(2)(3)(4) Note: 0! = 1 (by definition)

  17. Counting TechniquesRule of Combinations • You visit Baskin & Robbins. How many different possible 3 scoop combinations could you decide on for your ice cream cone if you select from their 31 flavors? • The total choices is n = 31, and we select x = 3. combinations

  18. Probability Distributions Binomial n! P(x) =π x(1-π) n-x n!(n-x)! π = probability of “event of interest” n = sample size (number of trials or observations) x = number of “events of interest” in sample for the number of heads in 3 coin flips, we would use… π = 0.5 and (1 – π) = 0.5 n = 4 X = can be 0, or 1, or 2, or 3 Each value of X will have a different probability Note: other sources may show this equation as or similar.

  19. Probability Distributions Binomial Example What is the probability of 2 successes in 7 observations if the probability of the event of interest is 0.4? x = 1, n = 7, and π = 0.4

  20. Probability Distributions Binomial Example The probability of a single customer purchasing an extended warranty is 0.35. What is the probability of having 3 of the next 10 customers purchase an extended warranty?

  21. Probability Distributions Using the Binomial Table P(x = 3) = .2522 Examples: n = 10, π = .35, x = 3 n = 10, π = .75, x = 2 P(x = 2) = .0004

  22. The shape of the binomial distribution… depends on the values of π and n Probability Distributions Shape of the Binomial n = 5 π = 0.5 n = 5 π = 0.1 P(X) P(X) .6 .6 .4 .4 X X .2 .2 0 1 2 3 4 1 5 0 2 3 4 5 0 0

  23. Binomial Distribution Characteristics • Mean • Variance and Standard Deviation Where n = sample size π = probability of the event of interest for any trial (1 – π) = probability of no event of interest for any trial

  24. Binomial Distribution Characteristics Examples n = 5 π = 0.1 P(X) P(X) n = 5 π = 0.5 .6 .6 .4 .4 X X .2 .2 0 1 2 3 4 1 5 0 2 3 4 5 0 0

  25. Binomial Distribution Characteristics Compound Events Individual probabilities can be added to obtain any desired combined event probability. • Examples: • The probability that less than 3 of the next 10 customers will purchase an extended warranty is • The probability that 3 or fewer of the next 10 customers will purchase an extended warranty is

  26. Binomial DistributionUsing Excel Cell Formulas n xπ n xπ x (1-π) x n π cum

  27. Probability Distributions Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Exponential Poisson Poisson Ch. 6 Ch. 7

  28. Probability Distributions The Poisson Distribution • Used when you are interested in… the number of times an event occurs within a continuous unit or interval. • Such as time, distance, volume, or area. • Examples: The number of… • potholes in mile of road (distance) • computer crashes in a day (time) • chocolate chips in a cookie (volume) • mosquito bites on a person (area) Was actually used in a study of malaria

  29. Probability Distributions The Poisson Distribution • Defining features: • Independent: The occurrence of any single event does not impact the occurrence of any other event. • Constant: The average probability of an event occurring in one area of opportunity remains the same for all other similar areas. The average number of events per unit is represented by (lambda) • The average probability of an event decreases or increases proportionally to any decrease or increase in the size of the area of opportunity.

  30. Probability Distributions The Poisson Distribution e -λλx P(x) = x! where: x = number of events in an area of opportunity  = expected value (mean) of number of events e = base of the natural logarithm system (2.71828...) Note: other sources may show this equation as or similar.

  31. Poisson Distribution Characteristics • Mean Isn’t that interesting? • Variance and Standard Deviation where  = expected number of events

  32. Probability Distributions Using the Poisson Table Example: Find P(X = 2) if  = 0.50

  33. Poisson DistributionUsing Excel Cell Formulas x λ cum

  34. Probability Distributions Graphing the Poisson  = 0.50 P(X=2) = 0.0758

  35. Probability Distributions Shape of the Poisson • The shape of the Poisson Distribution depends on the parameter  :  = 0.50  = 3.00

  36. Chapter Summary • Probability distributions • Discrete random variables • The Binomial distribution • The Poisson distribution

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