An Introduction to the Hypergeometric Distribution

An introduction to the hypergeometric distribution. I briefly discuss the difference between sampling with replacement and sampling without replacement. I describe the conditions required for the hypergeometric distribution to hold, discuss the formula, and work through 2 simple examples.

I also discuss the relationship between the binomial distribution and the hypergeometric distribution, and a rough guideline for when the binomial distribution can be used as a reasonable approximation to the hypergeometric. I finish with a brief example involving the multivariate hypergeometric distribution.

For those using R, here is the R code to find the probabilities for the examples in this video:

The probability of picking exactly 4 red balls when picking 5 balls from a source containing 6 red and 14 yellow.

Without replacement (hypergeometric):
choose(6,4)*choose(14,1)/choose(20,5)
[1] 0.01354489
or
dhyper(4,6,14,5)
[1] 0.01354489

With replacement (binomial):
dbinom(4,5,6/20)
[1] 0.02835

The probability of picking exactly 7 females when randomly sampling from a school with 1100 female and 900 male students.

Without replacement (hypergeometric):
choose(1100,7)*choose(900,3)/choose(2000,10)
[1] 0.1664901
or
dhyper(7,1100,900,10)
[1] 0.1664901

With replacement (binomial):
dbinom(7,10,1100/2000)
[1] 0.1664783

Multivariate hypergeometric, probability of picking exactly 3 Democrats, 2 Republicans, and 1 independent in the sample.

choose(12,3)*choose(24,2)*choose(8,1)/choose(44,6)
[1] 0.06881377
or, with the extraDistr package installed:
dmvrhyper(c(3,2,1),c(12,24,8),6)

[aymenechchalim]

Huge admiration for your work, it's nice to realize that something you did 9 years ago is still remarkably useful to many of us, thank you

[lazy_hiker_guy]

You are better at explaining the basic concepts of statistics than my college professors. You have become my shifu!

[yaribsuarez]

You're a gifted profesor, I really enjoy and understand!! 👍

[Cleisthenes]

Huh. So why is it 'hyper-geometric'?

[danielrojas]

your voice is so peaceful I dont know if Im studying or hearing music.
Great videos man

[narasimhann]

Dude!!
You're series is literally Gold.
Precise explanation, with examples and tiny bit of humor . Hats of to you!!!
I've learnt more now than during my college. Thank you :)

[kniix]

I absolutely love that you compare each dist type to Binomial, so that we can understand WHY they are different. Well done!

[sknight]

who knew a video from years ago would help me so much with my uni, hope you're still teaching because omg your explanation is so good.

[benjaminli]

I literally had a list of distributions i needed to review, and you just lined up all the videos, XD, amazing.

[orlandowan]

One of the better instructional videos on this topic here on YouTube. You also explain similarity and trade-off with the binomial distribution as well as cater for scenarios where there are more than two possible outcomes. Thanks.

[gabriel-braga-uc]

Your videos are serving as amazing insights and have been helping me a lot throughout my graduation. Thank you so much for this peak content!

[ozchelseagirl]

I am watching all your videos and I second, these are all relevant, clearly explained, brilliant videos. Thank you so much for your brilliant work!

[abhinavbichal]

You are the best statistics teacher I have ever found!

[Mariam-sonb]

I wish I found your videos a little bit earlier! Watching your videos give some sense of hope for my exam tomorrow. Thank you! :-)

[instant_mint]

These videos are helping me so much! I feel like I'm about to give up this course but then I come home and watch another video from this series, and suddenly things make sense. Thanks for making great videos

[bokangfalatsi]

Okay Professor, you are guilty of being the best Mathematical Statistics Prof.

[philandthai]

I am sure that it is a significant effort to create these videos, which are so smooth, so clear and so polished. I always think that statistics in an intrinsically difficult area of mathematics.While I am an advocate of book learning, your videos are a great supplement and I really appreciate your contribution.

[goksenumutguler]

Very clear and simplified version of everything teachers generally tend to tell complex as hell, you sir is a legend.

[nishadr.]

this is basically 5 hours of lectures summed up in 15 min

[umeshtiwari]

Hats off to you Sir, the way you explain really fits into the mind. Thanks