Hypergeometric_Distribution 044444.pptx

1. Hypergeometric
Probability Distribution
Group members:
K22ES004 K22ES10 K22ES12

2. Introduction to Hypergeometric Distribution
• The hypergeometric distribution is used to
determine the probability of a certain number of
"successes" in a series of draws made without
replacement from a fixed population. The
distribution depends on the size of the
population, the number of draws, and the
number of "successes" in the population.
• This distribution can be used as a model for
various scenarios which involve a series of
dependent trials that result in either a "success"
or a "failure".

3. • The formula for the hypergeometric distribution
is:
 Population (N):* The total number of items or objects.
 Successes (K):* The number of favorable outcomes
(the items we are interested in).
 Failures (N - K):* The remaining items in the
population that are not of interest.
 Sample size (n):* The number of items selected from
the population.
 Successes in the sample (x):* The number of
favorable outcomes found in the sample.

4.  Problem Statement:
In a jury selection process, there are 50 potential jurors, 25 of
whom are female. From these, 13 are randomly selected. We need
to determine:
1)The most likely number of female jurors selected.
2)The probability of selecting exactly 6 female jurors.
3)The probability of selecting more than 8 female jurors.

5. • Solution:
Key Information:
• Population size (N): 50 jurors
• Number of female jurors (K): 25 females
• Sample size (n): 13 jurors selected
• Random variable: Number of female jurors selected
• The hypergeometric distribution is used to model the probability of a
specific number of successes (female jurors) in a sample drawn from a
population, without replacement.
• 1. Most Likely Number of Female Jurors:
•To find the most likely number of female jurors, we can compute
the expected value:
•E(X)=
•For this problem:
•E(X)=​
=6.5
•Since the number of female jurors selected must be an integer, the
most likely number of female jurors selected is either 6 or 7.

6. 2. Probability of Selecting Exactly 6 Female Jurors:
•To find the probability of selecting exactly 6 female jurors, we
use the formula:
•P(X=6)=
•Breaking this down:
 represents the number of ways to select 6 female jurors from
the 25 available.
 represents the number of ways to select 7 non-female jurors
from the remaining 25.
 is the total number of ways to select 13 jurors from 50.
•After computing:
P(X = 6) approx 0.240 {or 24.0%})

7. 3. Probability of Selecting More than 8 Female Jurors
•The probability of selecting more than 8 female jurors is the sum
of the probabilities of selecting 9, 10, 11, 12, or 13 female jurors:
•P(X>8)=P(9)+P(10)+P(11)+P(12)+P(13)
•Using the same formula for each term, and summing them up:
•P(X > 8) approx 0.098 (or 9.8%)

8. HYPERGEOMETRIC DISTRIBUTION USES
• Hypergeometric distribution is used when the
probability of success in each trial changes as to find
the probability of successes.
• In acceptance sampling, hypergeometric distribution
plays a very important role and it is widely applied.