## Probability theory

I want to go through some prob­a­bil­i­ty the­o­ry with you before going on with dif­fer­ent dis­tri­b­u­tions of the data. When you are work­ing with dis­tri­b­u­tions, you are deal­ing with prob­a­bil­i­ties. I’ll elab­o­rate on this sub­ject under the bino­mi­al and multi­n­o­mi­al dis­tri­b­u­tion.

You don’t have to wor­ry; it is quite easy to work with prob­a­bil­i­ties. I’ll illus­trate this using the clas­si­cal heads or tails exam­ple. The prob­a­bil­i­ty of get­ting a head or tail at a coin toss is 0.5. But what if you want to cal­cu­late the prob­a­bil­i­ty of get­ting a head AND a head in two toss­es where the prob­a­bil­i­ty of a head in every sin­gle toss is 0.5. How to cal­cu­late this?

You can use a very sim­ple prin­ci­ple. You sim­ply mul­ti­ply the prob­a­bil­i­ty (p) of get­ting a head with get­ting a head; 0.5 x 0.5 = 0.52 = 0.25. The mag­ic word here is AND. AND means mul­ti­ply. Of course the prob­a­bil­i­ty has to be low­er than get­ting a Head once since there is a 50 % chance to get a Tail in the next toss. To do this, you mul­ti­ply to get the prob­a­bil­i­ty of get­ting the same out­come (e.g. Heads or Tails) a spe­cif­ic num­ber in a row. This can be expressed pn, where p is the prob­a­bil­i­ty of the out­come (Heads or Tails) and n is the num­ber of times in a row you get this out­come as a result.

But, how a about get­ting a Head OR a Tail in every sin­gle toss? It is very sim­ple when you think about it; it is the sum of the prob­a­bil­i­ty of get­ting a Head and a Tail, which is 0.5 + 0.5 = 1. The prob­a­bil­i­ty of get­ting either a Head OR a Tail is 1 or 100 %. Right? So when the out­come does not mat­ter, or when you cal­cu­late the prob­a­bil­i­ty of get­ting sev­er­al types of out­comes, you add the prob­a­bil­i­ties. The mag­ic word in this case is OR.

Exam­ple 1

You are toss­ing a dice and want to cal­cu­late the prob­a­bil­i­ty of get­ting a 2 five times in a row. That is a 2 AND a 2 AND a 2, etc.

First, you need to cal­cu­late the prob­a­bil­i­ty of get­ting a 2. A dice has six sides. So the prob­a­bil­i­ty of get­ting one of the sides is $\frac{1}{6}$ or about 0.17. So the prob­a­bil­i­ty of get­ting a 2 is $\frac{1}{6}$.

Sec­ond, to cal­cu­late the prob­a­bil­i­ty of get­ting a side with a 2 five times in a row, you mul­ti­ply the prob­a­bil­i­ty of get­ting a two in a sin­gle toss ( $\frac{1}{6}$ ) by 5 (the num­ber of toss­es.) That is $(\frac{1}{6})^5$, which equals 0.0001.

Exam­ple 2

What is the prob­a­bil­i­ty of get­ting more than 4 in a sin­gle toss with the dice? That is a 5 OR six.

First you need to cal­cu­late the prob­a­bil­i­ty of get­ting one of the sides. That is ( $\frac{1}{6}$ ) as in the pre­vi­ous exam­ple.

Then sum the prob­a­bil­i­ties of the sides of the side with more than 4. That is two sides, the ones with 5 and 6. The result is $\frac{1}{6}+\frac{1}{6}=\frac{2}{6} = \frac{1}{3}$ or 0.33.

Impor­tant to remem­ber

(1) When you cal­cu­late the prob­a­bil­i­ty of the same out­come sev­er­al times in a row (Heads AND Heads) you mul­ti­ply.

(2) When you cal­cu­late the prob­a­bil­i­ty of dif­fer­ent out­comes at the same event (or toss), Heads OR Tails) you add.