![Hypergeometric
Distribution
Example4:Suppose that a shipment contains
.If 7 items are selected at random without replacement , what is
the probability that at least 3 defective items will be obtained?
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Defective Items Non Defective Item Total
5 10 15](https://image.slidesharecdn.com/hypergeometricdistribution-160325140617/85/Hypergeometric-distribution-13-320.jpg)
Hypergeometric distribution
- 2. 2 Hypergeometric Distribution Formula Definition In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes in draws, without replacement.
- 3. Hypergeometric Distribution Mean And Variance Hypergeometric Distribution Binomial Distribution N S N nN N Sn XVar −⋅ − − ⋅ ⋅ = 1 1 )( N sn XE ⋅ == )(µ )1)()(()( ppnXVar −=
- 4. 4 Hypergeometric Distribution characteristics The hypergeometric distribution has the following characteristics: There are only 2 possible outcomes. The probability of a success is not the same on each trial without replacement, thus events are not independent In which population is finite . Trials are dependent
- 5. Hypergeometric Distribution It is similar to the binomial distribution. But the difference is the method of sampling Binomial experiment: Sampling with replacement Hypergeometric experiment: Sampling without replacement
- 6. 6 Hypergeometric Distribution Example 1 In PlayTime Toys, Inc., Five employees are selected at random to form a committee to meet with management regarding shift starting times. What is the probability that four of the five selected for the committee belong to a union? Union Employee 40 Not Union Employee 10 Total Employee 50
- 8. Hypergeometric Distribution Example 2 :A carton contains light bulbs What is the probability that, if a sample of six is chosen at random from the carton of bulbs, x will be defective? ( ) ( ) ( )24 6 21 6 3 )( xx xXP −⋅ == ( ) ( ) ( ) 40316.0)0( 24 6 21 6 3 0 = ⋅ ==XP That is no defective Defective 3 Total 24
- 9. Hypergeometric Distribution (example continued) ( ) ( ) ( ) 00988.0)3( 24 6 21 3 3 3 = ⋅ ==XP That is 3 will be defective.
- 10. EXAMPLE 3 In a bag containing select 2 chips one after the other without replacement. Hypergeometric Distribution Red Chips 7 Blue Chips 5 Total Chips 12
- 11. Multivariate Hypergeometric Distribution X1 , X2 , X3 : Joint Probability Function
- 12. (1) Joint probability function: 5 2 3 z y x (2) In a box, there are 3 red balls, 2 blue balls, and 5 yellow balls. You select 4 balls. (1) Joint probability function for X, Y, and Z (2) Probability to select 1 red ball, 1 blue ball, and 2 yellow balls.
- 13. Hypergeometric Distribution Example4:Suppose that a shipment contains .If 7 items are selected at random without replacement , what is the probability that at least 3 defective items will be obtained? =≤−=≥ )2(1)3( XPXP [ ] 4267.0)2()1()0(1 =++− PPP ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3916.0)2( 1631.0)1( 0186.0)0( 15 7 10 5 5 2 15 7 10 6 5 1 15 7 10 7 5 0 = ⋅ = = ⋅ = = ⋅ = P P P Defective Items Non Defective Item Total 5 10 15
- 14. EXAMPLE 5 4 3 10 4 5 3 10 5 4 3 6 2 10 5 ( )( ) (3) ( )( ) 4(15) .238 252 C C P C C C C − − = = = = The National Air Safety Board has Safety Board will only be able to investigate five of the violations. What is the probability that three of five violations randomly selected to be investigated are actually violations? List Reported Safety Violation 10 Actual Violation 4 Hypergeometric Distribution
- 15. 15 Hypergeometric Distribution Example 1 In PlayTime Toys, Inc., Five employees are selected at random to form a committee to meet with management regarding shift starting times. What is the probability that four of the five selected for the committee belong to a union? Union Employee 40 Not Union Employee 10 Total Employee 50
- 17. Any Question…?