In the fields of statistics and probability theory, a hypergeometric distribution is essentially a distinct probability distribution that explains the probability of k successes (i.e., some random draws for the thing drawn that has some specified feature) in n number of attractions, without any replacement, from a given population size N that includes exactly K objects having that feature, where the interest may succeed, or it may fail.
In other words, the hypergeometric distribution is a probability distribution that defines the probability of k successes.
It is possible to derive the formula for the probability of a hypergeometric distribution by first determining the number of items in the population, the number of items in the sample, the number of successes in the people, the number of hits in the model, and a small number of possible combinations.
A mathematical representation of the probability looks like this:
P = KCK * (N-K) C (n-k) / N Cn
N = Total No. of items in the population
N = Total No. of things in the sample
K = No. of success in the population
K = No. Of success in the sample
Firstly, calculate the total number of objects in the population, which N indicates. As an illustration, there are 52 playing cards in a standard deck.
After that, count the number of things included in the sample, which the letter n will symbolize. For instance, count the number of cards taken from the deck.
Then, define the occurrences in the population that will be judged to be successful, and we will refer to this as K. For example, the number of hearts in the whole deck is 13.
Next, identify the occurrences that will be deemed successes in the selected sample; we will designate this variable as k. For example, the number of hearts that appeared on the cards taken from the deck.
The last step in deriving the formula for the probability of a hypergeometric distribution is to use the number of successes in the population (step 3), the number of wins in the sample (step 4), the number of items in the population (step 1), and the number of items in the model (step 2). The formula for the chance of a hypergeometric distribution is mentioned above.
Real-life Examples of Hypergeometric distribution
Playing Poker
Imagine having access to a standard deck of playing cards, and the game rules require you to draw five cards simultaneously. With the hypergeometric distribution, it is simple to calculate the likelihood that every card drawn is a spade.
This probability may be used in a variety of games. It is essential to highlight that the binomial distribution cannot be utilized in this situation since the cards are drawn without replacement. This implies that the likelihood that the experiment will be successful shifts with each new draw that is conducted.
The Number of Voters
Imagine that there are 100 female and 200 male voters in a constituency. The hypergeometric probability distribution may be used to determine the likelihood that eight of those voters will be male, given a sample size of ten voters chosen randomly. This probability can be computed using the hypergeometric probability distribution.
Candy Boxes
Imagine you have a box of sweets with ten candies in total; six candies have a sweet flavor, while the other four have a sour taste. With the assistance of a geometric distribution function, you can calculate the likelihood that three candies will have a sweet flavor. In contrast, the remaining one will taste sour when you choose four sweets randomly from the box.
Selecting Committee Members
Let’s say that to successfully arrange an annual event, an educational institution like a school or college requires the help of a diverse team comprised of administrators, students, professors, and lab assistants.
Let’s imagine that out of the whole school population, there are 60 instructors, 100 pupils, 30 lab assistants, and ten principals, and twenty persons are chosen randomly. The geometric probability distribution makes it possible to quickly calculate the probability that the newly formed committee will consist of two teachers, four students, three lab assistants, and one principal.
This is likely thanks to the fact that all of these variables can be represented by triangles.
Color Balls
Let’s say there are ten green balls, twenty red balls, ten yellow balls, and five white balls inside a box. Let’s assume ten balls are picked randomly from the box; with a hypergeometric probability distribution, we can determine the likelihood that three will be blue.
A similar process may be used to calculate the likelihood that, for example, four will have the color yellow, two will have a shade of white, two will have the color red, etc.
Rolling More Than Two Dies
Rolling many dice at the same time is an illustration of a hypergeometric distribution, which is one of the most famous instances. If we roll six dice simultaneously, we can use the hypergeometric distribution to estimate the probability that four dice will have an even number on their top face. In contrast, two dice will have an odd number on their faces.
This probability can be estimated by rolling six dies simultaneously.
Finding Defective Products
Imagine that a shoe factory has been tasked with producing two hundred pairs of shoes. When the shipment is packed, ten shoe boxes will be chosen randomly for inspection to search for flaws and errors.
It is necessary to compute the likelihood that six out of the ten different pairs of shoes will have some flaw. In this kind of scenario, the hypergeometric distribution is the kind of probability distribution that works the best.
Students in Various Disciplines
Imagine that a total of 2000 kids are enrolled in a particular school. One hundred students are chosen at random to create a student organization dedicated to community service. It is necessary to compute the likelihood that fifty selected students come from the arts and humanities discipline, twenty come from the science department, and the remaining thirty come from the business department.
Since the number of students who are participating in the experiment as a whole as well as the size of the sample that has been selected, is both limited, and since there will be no replacement of models, the hypergeometric probability distribution is the one that should be used in this scenario.
Mobile Repairing Shops
Imagine that a mobile phone repair business gets twenty mobile phones that must be fixed to fulfill their customers’ needs daily. Let’s say that the next day, the proprietor of the mobile phone repair business chooses ten mobile phones at random and hands them over to the employee who works there.
The hypergeometric probability distribution is used to compute the likelihood that six of the chosen mobile phones will have a problem with their hardware. Still, only four of them will have a problem with their software.
This makes it easier for the proprietor of the business to maintain spare parts on hand if it becomes essential to repair components found inside the mobile phone’s internal circuitry.
Getting a Good Job
The hypergeometric probability distribution function helps determine the likelihood of being chosen for a specific internship. You may use this function to determine your chances of getting the position.
Imagine that there are a total of 25 persons interested in participating in the internship program. With hypergeometric probability distribution, one can compute the likelihood that the first twenty applications will be picked for the personal interview.
With the assistance of hypergeometric distribution, it is also possible to express in a like fashion the likelihood of ten applicants qualifying for the personal interview out of the twenty that were picked as candidates.