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HYPERGEOMETRIC DISTRIBUTION Dependent Trials

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Learning Goals I can use terminology such as probability distribution, random variable, relative frequency distribution, etc. I can give examples of situations where a hypergeometric distribution exists. I can list the criteria that makes for a hypergeometric distribution. I can state and use the formula for a hypergeometric distribution. I can state and use the expected value formula for a hypergeometric distribution.

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Why another distribution? When choosing the starting line-up for a game, or dealing cards from a standard deck, there can be no repetitions. In these situations, each selection reduces the number of items that could be selected in the next trial. Thus the probabilities in these trails are dependent! As a result, we cannot use a Binomial Distribution!

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Investigation In Ontario, a civil-court jury requires only 6 members. Suppose a civil-court jury is being selected from a pool of 18 citizens, 8 of whom are men. a) Determine the probability distribution for the number of women on a civil court jury b) What is the expected number of women on the jury?

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Probability Distribution Table XP(x)

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Expected Value

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Conditions for Hypergeometric There are n dependent trials There are only two possible outcomes for each trial - success or failure

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HyperGeometric Distribution

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Hypergeometric Distribution Short version that might be easier to remember: X is a discrete random variable corresponding to the number of` successes; a is the total number of possible successes (eg. Total number of women available, etc.); k is the number of successes; & r is the number of items you want in total (eg. Total number of people on the committee).

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Example 1 Suppose that we randomly select 5 cards without replacement from a deck of 52 cards. a) Describe how this meets the conditions of a hypergeometric distribution. b) Determine the expected number of red cards.

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Notice this… Probability Distribution: XP(X)

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Example 2 A box contains 7 yellow, 3 green, 5 purple, and 6 red candies jumbled together. What is the expected number of red candies among 5 candies taken randomly from the box?

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Example 3 In a group of 10 people, 6 have white shoes and the rest have black shoes. If 4 people are chosen at random, a) What is the probability 3 have white shoes? b) What is the expected number of white shoes?

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Homework See worksheets

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