{"id":16342,"date":"2016-06-01T13:10:09","date_gmt":"2016-06-01T13:10:09","guid":{"rendered":"https:\/\/ygorganization.com\/?p=16342"},"modified":"2016-06-01T13:10:09","modified_gmt":"2016-06-01T13:10:09","slug":"maths2","status":"publish","type":"post","link":"https:\/\/ygorganization.com\/staging\/?p=16342","title":{"rendered":"Demystifying Probability, Part 2"},"content":{"rendered":"

In the previous article, I introduced Rustywolf\u2019s <\/span>Probability Calculator<\/span><\/a> and we saw the calculator in action. In this article, we delve into the maths behind it.<\/span><\/p>\n

<\/p>\n

Binomial Coefficients
\n<\/b>First, we need to understand what a Binomial Coefficient is. Don\u2019t be put off by the fancy mathematical term, it\u2019s a very concrete and easy to understand object. <\/span><\/p>\n

A binomial coefficient is \u201cthe number of ways we can pick things out of a group.\u201d They look like this:<\/span><\/p>\n

\"CodeCogsEqn<\/p>\n

You read it as \u201c4 choose 2\u201d. And that\u2019s exactly what it means: it\u2019s the number of ways of choosing 2 things out of a group of 4.<\/span><\/p>\n

What does this look like? Let\u2019s take 4 cards, A, B, C, D. Let\u2019s write out the number of different combinations of 2 cards there are.<\/span><\/p>\n

AB
\n<\/span>AC
\n<\/span>AD
\nBC
\n<\/span>BD
\n<\/span>CD<\/span><\/p>\n

That\u2019s all of the possible combinations. Any other combination would just be one of the above, but maybe in a different order. So \u201c4 choose 2\u201d = 6, because there are 6 ways of choosing 2 cards out of\u00a04.<\/span><\/p>\n

There is a mathematical formula for calculating the value of a binomial coefficient, but we won\u2019t need\u00a0it. If we want to calculate any, we can ask Google directly:<\/span><\/p>\n

http:\/\/prnt.sc\/b713x3<\/span><\/a><\/p>\n

Google is great. It makes a convenient calculator as we can type any equations we need\u00a0directly into it.<\/p>\n

Similarly we can calculate the number of possible 5-card hands of a 40-card deck. We want the value of “40 choose 5”, or in maths notation,\u00a0\"CodeCogsEqn\"\u00a0:<\/span><\/p>\n

http:\/\/prnt.sc\/ba8325<\/a><\/p>\n

Of course, if we\u2019re running multiple copies of a particular card, some of these hands would look the same.<\/span><\/p>\n

The Hypergeometric Distribution
\n<\/b>\u2018Hypergeometric Distribution\u2019 is, as far as we’re concerned, just a fancy mathematical term for the maths describing drawing cards from a deck. <\/span><\/p>\n

It\u2019ll be easiest to see as an example. Let\u2019s suppose we want to know the probability in a 5-card hand of drawing the 1 Allure we run, but no Darks out of 5, out of a 40 card deck in total. <\/span><\/p>\n

This means:<\/span><\/p>\n