Hypergeometric probability distribution

Hypergeometric probability distribution

1. NADEEM UDDIN ASSOCIATE PROFESSOR OF STATISTICS HYPERGEOMETRIC PROBABILITY DISTRIBUTION

2. Hypergeometric Experiments : A hypergeometric experiment is a random experiment that has the following properties: 1-The experiment consists of n repeated trials . 2-Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. 3-The probability of success, in each trial is not constant . 4-The trials are dependent.

3. Experiment-1 A recent study found that four out of nine houses were insured. If we select three houses from the nine without replacement and all the three are insured. Explanation : 1-The experiment consists of 3 repeated trials . 2-Each trial can result in just two possible outcomes. We call one of these outcomes a success (insured) and the other, a failure ( not insured ). 3-The probability of success, in each trial is not constant. 1 st trial probability would be . 2 nd trial probability would be . 3 rd trial probability would be . 4-The trials are dependent.  

4. Experiment-2 You have an urn of 15 balls - 5 red and 10 green. You randomly select 2 balls without replacement and count the number of red balls you have selected. Explanation: This would be a hypergeometric experiment. Note that it would not be a binomial experiment. A binomial experiment requires that the probability of success be constant on every trial. With the above experiment, the probability of a success changes on every trial. In the beginning, the probability of selecting a red ball is 5/15. If you select a red ball on the first trial, the probability of selecting a red ball on the second trial is 4/14. And if you select a green ball on the first trial, the probability of selecting a red ball on the second trial is 5/14. Note further that if you selected the balls with replacement, the probability of success would not change. It would be 5/15 on every trial. Then, this would be a binomial experiment.

5. Hypergeometric Random Variable : A hypergeometric random variable is the number of successes that result from a hypergeometric experiment . Hypergeometric Distribution : The probability distribution of a hypergeometric random variable is called a hypergeometric distribution. Given x , N , n , and k , we can compute the hypergeometric probability based on the following formula: The mean of the distribution is equal to n × k / N . The variance is n × k × ( N - k ) × ( N - n ) / [ N 2 × ( N - 1 ) ] .  

6. Where N = Size of population . n = Size of sample . K = Number of successes . N – K = Number of failures . x = Number of successes in the sample .

7. Use of HyperGeometric Probability Distribution   Example-1 If 5 cards are dealt from a deck of 52 playing cards, what is the probability that 3 will be hearts?   Solution: By using hypergeometric distribution Where N = 52 n = 5 K = 13 P(x=3) = ?  

8. Example-2 From 15 kidney transplant operation,3 are fail within a year .consider a sample of 2 patients , find the probability that (a)Only 1 of the kidney transplant operations result in failure within a year . (b)All 2 of the kidney transplant operations result in failure within a year . (c)At least 1 of the kidney transplant operations result in failure within a year.

9. Solution: By using hypergeometric distribution Where N =1 5 n = 3 K = 3 (a) P(x=1 ) = ? (b) P(x=2 ) = ?  

10. (c) P ( x ≥ 1) = ? P(x ≥ 1) = 0.4351 + 0.0791 (from part a and b ) P(x ≥ 1) = 0.5142  

11. Example-3 A small voting district has 1000 female voters and 4000 male voters. A random sample of 10 voters is drawn. What is the probability exactly 7 of the voters will be male?   Solution: Since the population size is large relative to the sample size, We shall approximate the hypergeometric distribution to binomial distribution. Therefore n = 10 q = 0.2(female probability)