Hypergeometric Distribution Help

What do these properties mean?

• Population and Sample

  The population is where all of the objects reside before you have begun your experiment. The sample is where every object goes once you have drawn it from the population.

• Without Replacement

  This means that when taking draws out of the population, each object is not replaced in the population after it is selected. This means that whenever an object is drawn, every draw afterwards is affected since it no longer has the chance to draw the previously selected object.

• Without Regard to Order

  This means that the order that the objects are drawn in has no affect on the outcome of the experiment. The only thing that is counted is the number of objects that have the trait that is considered a success - the order that they were drawn in is not taken into consideration.

• Trait that is Considered a Success

  When conducting a hypergeometric experiment, out of the population, there are a certain number of them that have some trait that is of interest. This is the trait that is considered a success - and out of the sample of objects you draw from the population, the number that have this trait is the random variable. Because this is the random variable, it means that it is not determined beforehand - the number of objects drawn which have the trait considered a success is likely to change from experiment to experiment.

Example

When dealing with an experiment, if you see or hear the words "without replacement" you are probably dealing with a hypergeometric experiment.

For example, imagine you have a standard deck of 52 cards and you are interested in seeing how many out of a hand of five cards are a face card (Jacks, Queens, and Kings). You draw five random cards from the population, which is the deck of cards, and the cards that you draw become your sample, which is the hand. Each time you select a card, you put it into your hand and do not replace it back in the deck the next time you draw another card. This would be a hypergeometric distribution with N = 52 (because there are 52 cards in the deck), r = 12 (because there are 12 face cards in the deck), and n = 5 (because you are drawing a hand of five cards). The only thing left would be to determine how many of those five cards you would like to be face cards, and that would be x (because it is the number of sample successes that you would like). Note that you will not always get exactly that number of successes - there is only some probability that you will.

Requirements

• The number of objects in the population must be a whole number greater than zero and at least equal to the number of objects in the population which have the trait that is considered a success. Also, the number must be at least equal to the number of objects in your sample (N >= r, N >= n, and N = 1, 2, 3 ...).
• The number of objects in the population that have the trait that is considered a success must be a whole number greater than or equal to zero and also at least equal to the number in your sample that have the trait that is considered a success (r >= x and r = 0, 1, 2 ... N).
• The number of objects in your sample must be a whole number greater than zero and at least equal to the number of objects in your sample that have the trait that is considered a success (n >= x and n = 1, 2, 3 ... N).
• The number of objects in your sample that have the trait that is considered a success must be a whole number greater than or equal to zero (x = 0, 1, 2 ... lower of r and n).