1 / 21

Hypergeometric Distributions

Hypergeometric Distributions. When choosing the starting line-up for a game, a coach has to select a different player for each starting position – obviously!

dennis-leach
Download Presentation

Hypergeometric Distributions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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.

  2. Hypergeometric Distributions • Recall, the concepts associated with a Binomial Distribution: • nidentical 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.

  3. 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

  4. 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

  5. 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?

  6. Hypergeometric Distributions Sol’n • 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…

  7. 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…

  8. Hypergeometric Distributions Cont… • Probability of a jury with x women is…

  9. Hypergeometric Distributions Cont… • Using the formula for the probability…

  10. Hypergeometric Distributions Cont… Graphically…

  11. Hypergeometric Distributions Cont… • Expected Value…

  12. Hypergeometric Distributions • Generalizing the methods lead to… Probability in a Hypergeometric Distribution The probability of xsuccesses in r dependent trials -where: n population size & a is the number of successes in the population

  13. 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)

  14. 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 Sol’n • n=7+3+5+6=21(# of candies in box {popl’n}) r=5 (# of candies removed {trials}) a=6 (# of red candies {successes})

  16. Hypergeometric Distributions Con’t… • Using the general expectation formula

  17. 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.

  18. Hypergeometric Distributions • 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.

  19. 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. r=40 (# raccoons caught in summer {trials}) a=500 (# tagged raccoons {population})

  20. Hypergeometric Distributions • So, • Therefore, the # of raccoons in the area is about 1333.

  21. Hypergeometric Distributions Probability in a Hypergeometric Distribution The probability of xsuccesses in r dependent trials is: -where: n population size & a is the number of successes in the population

More Related