Hypergeometric Distributions

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Hypergeometric Distributions

• When choosing the starting line-up for a game, a
  coach has to select a different player for each
  starting position obviously!
• Similarly, when a person is elected to represent
  student council or you are dealt a card from a
  standard deck, there can be no repetition.
• In such situations, each selection reduces the
  number of items that could be selected in the
  next trial.
• Thus, the probabilities in these trials are
  dependent
• Often, we need to compute the probability of a
  specific number of successes in a given number of
  dependent trials.

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Hypergeometric Distributions

• Recall, the concepts associated with a Binomial
  Distribution
• n identical trials (Bernouli Trials)
• Two possible outcomes (Success or Failure)
• Probability of Success does not change with each
  trial
• Trials are independent of one another
• Purpose is to determine how many successes occur
  in n trials.

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Hypergeometric Distributions

• With Hypergeometric Distributions, some of the
  concepts are transferable
• Each trial has possibility only for success or
  failure
• Specific number of trials
• Random variable is the number of successful
  outcomes from the specified number of trials
• Individual outcomes cannot be repeated within the
  trials

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Hypergeometric Distributions

• There are also some differences
• Each trial is dependent on the previous trial
• The probability of success changes with each
  trial
• Calculations of probabilities in Hypergeometric
  Distributions generally require formulae using
  combinations

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Hypergeometric Distributions

• Example 1
• Civil trials in Ontario require 6 jury members.
  Suppose a civil-court jury is being selected from
  a pool of 18 citizens, 8 of whom are men.
• Determine the probability distribution for the
  number of women on the jury.
• What is the expected number of women on the jury?

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Hypergeometric Distributions

• Soln
• Selection process involves dependent events since
  each person who gets chosen cannot be selected
  again.
• Look to combinations for total number of ways 6
  jurors can be selected from the pool of 18

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Hypergeometric Distributions

• Cont
• There can be between 0 and 6 women on the jury.
  The number of ways in which x women can be
  selected is 10Cx.
• Thus, the men can fill the remaining 6-x
  positions on the jury in 8C6-x ways.
• Therefore, the number of ways of selecting a jury
  with x women on it is the product of the two

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Hypergeometric Distributions

• Cont
• Probability of a jury with x women is

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Hypergeometric Distributions

• Cont
• Using the formula for the probability

of women x Probability P(x)
0 0.00151
1 0.03017
2 0.16968
3 0.36199
4 0.31674
5 0.10860
6 0.01131
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Hypergeometric Distributions

• Cont
• Graphically

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Hypergeometric Distributions

• Cont
• Expected Value

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Hypergeometric Distributions

• Generalizing the methods lead to
• Probability in a Hypergeometric Distribution
• The probability of x successes in r dependent
  trials
• -where n population size a is the number of
  successes in the population

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Hypergeometric Distributions

• Expected Value for a Hypergeometric Distribution
• Notes
• Ensure that the number of trials is
  representative of the situation
• Each trial is dependent (no replacement between
  trials)

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Hypergeometric Distributions

• Example 2
• A box contains seven yellow, three green, five
  purple, and six red candies jumbled together.
• What is the expected number of red candies among
  five candies poured from the box?
• Verify that the expected value formula for H.D.
  gives the same value as the expectation for any
  probability distribution.

15
Hypergeometric Distributions

• Soln
• n735621( of candies in box popln)
• r5 ( of candies removed trials)
• a6 ( of red candies successes)

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Hypergeometric Distributions

• Cont
• Using the general expectation formula

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Hypergeometric Distributions

• Example 3
• In wildlife management, the MoE caught and tagged
  500 raccoons in a wilderness area. The raccoons
  were released after being vaccinated against
  rabies. To estimate the raccoon population in
  the area, the ministry caught 40 raccoons during
  the summer. Of these, 15 had tags.
• Discuss why this can be modeled with a
  hypergeometric distribution.
• Estimate the raccoon population in the area.

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Hypergeometric Distributions

1. The 40 raccoons captured in the summer were all
   different from each other. In other words, there
   were no repetitions, thus the trials were
   dependent. The captured raccoon was either
   tagged (success) or not (failure). Therefore,
   the situation has the characteristics of a
   hypergeometric distribution.

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Hypergeometric Distributions

• Assume that the number of tagged raccoons caught
  in the summer is equal to the expected number of
  raccoons for the hypergeometric distribution.
  Substitute the the known values into the formula
  and solve for the population size, n.
• r40 ( raccoons caught in summer trials)
• a500 ( tagged raccoons population)

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Hypergeometric Distributions

• So,
• Therefore, the of raccoons in the area is about
  1333.

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Hypergeometric Distributions

• Probability in a Hypergeometric Distribution
• The probability of x successes in r dependent
  trials is
• -where n population size a is the number of
  successes in the population