SDS LA-1

Abstract

Discrete probability distribution
Discrete probability distribution is a type of probability distribution that shows all possible values of a discrete random variable such as 0, 1, 2, 3… along with the associated probabilities. Examples of discrete probability distributions:

1. Binomial Distribution

2. Poisson Distribution

3. Bernoulli Distribution
   

Hypergeometric distribution
Hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure.[1]

Introduction

Hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure. The hypergeometric distribution can be used for sampling problems such as the chance of picking a defective part from a box (without returning parts to the box for the next trial).

Uses

1. • The distribution is commonly studied in most introductory probability courses.

   • We also use this distribution to estimate the number of fishes in the lake.

Probability Mass Function

Parameters required:
1. Number of successes
2. Number of trials
3. Population size

Conditions for hypergeometric distribution:
1. Total number of items (population) is fixed.
2. Sample size (number of trials) is a portion of the population.
3. Probability of success changes after each trial.

Formulae

Formula to calculate the probability of a random variable:

where,
K is the number of successes in the population k is the number of observed successes N is the population size
n is the number of draws

Mean and Variance

Mean & Variance of hypergeometric distribution

where,

N is the total population size

m is the number of defective items

n is the sample size

Real life applications of distribution

Some real life applications are:

1. Poker
   Suppose you have a fair deck of playing cards, and you are supposed to draw five cards at a time. The probability that all the cards that are drawn are spades can be calculated easily with the help of hypergeometric distribution.
   
   

   Number of Voters
   Suppose that a district consists of 100 female voters and 200 male voters. If a group of ten voters is selected at random, then the probability that eight of the selected voters would be male can be calculated with the help of hypergeometric probability distribution.
   
   

   Rolling Multiple Dies
   One of the prominent examples of a hypergeometric distribution is rolling multiple dies at the same time. Suppose six dies are rolled simultaneously, then the probability that four of the dies would have an even number on their top face, while two dies would have an odd number on the top, can be estimated with the help of hypergeometric distribution.[2]
   
   

Problems and solutions related to distribution

Q1: The following problems involve being dealt, at random and without replacement, a bridge hand (i.e., 13 cards) from a standard deck of 52 cards. Find the probability of being dealt a bridge hand that:

1. does not contain a spade?

2. contains exactly 5 hearts?

3. contains at most one ace?

Solution:

1. (39C13)(52C13)≐0.0128(39C13)(52C13) = 0.0128

2. (13C5)(39C8)(52C13)≐0.1247(13C5)(39C8)(52C13) = 0.1247

3. P(no aces)+P(one ace)=(48C13)(52C13)+(48C12)(4C1)(52C13) = 0.7427

Q2: In a small pond there are 50 fish, 10 of which have been tagged. A fisherman’s catch consists of 7 fish (assume his catch is a random selection done without replacement). What is the probability that exactly 2 tagged fish are caught?[3]

Solution: P(X=2)=(10C2)(40C5)(50C7)=0.296
`

Conclusion

The hypergeometric distribution is one of the most useful probability distributions which can be used for discrete probability estimations. It is a distribution which finds probability of success of k out of n ways in a sequence of experiments without replacement. Its different from binomial as binomial is applied for with replacement condition.

Reference

[1]. https://en.wikipedia.org/wiki/Hypergeometric_distribution

[2]. https://studiousguy.com/hypergeometric-distribution-examples/

[3]. http://mathcenter.oxford.emory.edu/site/math117/probSetHypergeometricDistribution/