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Probability Distributions and the Binomial Distribution BUSA 2100, Sections 5.1 - 5.4

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Random Variables l A random variable is a quantity that changes values from one occurrence to another, in no particular pattern. l Example 1: Let X = number of heads on 3 tosses of a coin; X = 0, 1, 2, 3. l Example 2: Let Y = number of papers sold daily in a newspaper rack; Y = 0, 1, 2, 3, 4, 5.

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Discrete and Continuous Random Variables l A discrete random variable can have only a small number of specific possible values (usually whole numbers). It tells “how many.” The random variables in Examples 1, 2 are discrete. l A continuous random variable can have an undetermined large finite number of values. It tells “how much.”

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Continuous Random Variables l Example 3: Let W = amount of time in minutes that a family watches TV in a month. This is a continuous random variable. l Example 4: Weight and distance are also continuous random variables.

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Probability Distributions l A probability distribution (for a discrete random variable) is a list of all possible outcomes together with their associated probabilities. l State probability distribution for 3 coins.

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Expected Value l What is the “average” value for X for the probability XP(X) distribution 60.1 shown at70.2 the right? 80.3 90.4 l The median is 75; the simple average is 75; is there a more accurate answer?

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Expected Value, Page 2 l.l.

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Republican/Democrat Ex. l Ex. 1A: A large group of people consists of 40% Republicans and 60% Demo- crats. If a sample of 5 is chosen, what is the probability of getting 2 Republicans and 3 Democrats, in that order?

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Rep/Dem Example, Page 2 l Ex. 1B: What is the prob. of choosing 2 Repub. and 3 Democrats in any order?

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Rep/Dem Example, Page 3 l To avoid listing the number of ways that 2 R’s and 3 D’s can be arranged, we can use combinations to determine how many ways we can choose which two of the 5 selections will be Republicans. l The Republican/Democrat problem is an example of the binomial distribution.

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Intro. to Binomial Distribution l Binomial problems have 2 characteristics (requirements). l (1) Most important characteristic: Each selection must have exactly two possible outcomes. l Examples: l (2) Each selection is independent of the other selections.

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Binomial Notation & Formula l Do each item for Rep./Dem. problem. l n = number of selections l Define “success”. l p = probability of success on one selection l r = number of successes

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Binomial Problems, Page 1 l Example 1: A large lot of manufactured items contains 10% defectives. In a random sample of 6 items, what is the probability that exactly 2 are defective? l

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Binomial Problems, Page 2 l Example 2: Thirty percent of customers that enter an appliance store make pur- chases. What is the prob. that 4 of the next 10 customers will buy something?

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Binomial Problems, Page 3 l For convenience we will use a binomial table, looking up n, p, r, in that order. l Example 3A: A large lot of manufac- tured items contains 20% defectives. In a random sample of 8 items, what is the probability that 5 or more items are defective?

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Binomial Problems, Page 4 l Ex. 3B: P(2 or fewer) =

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Binomial Problems, Page 5 l Example 4: Sixty percent of the workers in a plant belong to a union. A random sample of 12 is chosen. Find the prob- ability that exactly 4 belong to a union.

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Binomial Problems, Page 6 l.l.

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Binomial Problems, Page 7 l Example 5: At Blaylock Company, in the past, 25% of new employees were not hired for a permanent position after a six-months probationary period. Among 7 new employees, what is the prob. that 5 or more will be hired permanently?

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Binomial Problems, Page 8 l It is essential that the values of p and r are consistent with the way that success is defined.

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