4.2 Probability - definition

To me probability is one of the key concepts in statistics, after all any statistical software will gladly calculate the famous p-value (a form of probability) for you. Still, let’s get back to our probability definition (see the sub-chapter name).

As said, at the conclusion of the previous section (Section 4), probability is a way to measure certainty. It’s like with the grades in school. In Poland a pupil can score 1 to 6 (lowest to highest grade) and this tells us how well he mastered the subject. If I score 1 then I didn’t master it at all, but when I get 6 this means that I got it all. We know from everyday life that probability takes values from 0 to 100%, e.g.

• Are you sure of it?
• Absolutely, one hundred percent.

or

• Do you think he can make it?
• I would say it’s fifty-fifty.

or even

• What are the chances?
• Pretty much, zero.

When something is bound to happen we assign it the probability of 100%.

When it can go either way we say fifty-fifty (50% it will happen, 50% it will not happen).

When an event is impossible we say zero (probability of it happening is 0%).

And this is the way statisticians use it. OK, maybe not quite. A typical statistics textbook will say that the probability takes values from 0 to 1. It is expressed this way for a few particular reasons (some of the reasons may be given later). Moreover, believe it or not, but it is actually compatible with our understanding that is based on everyday life.

From primary school (see also Wikipedia’s definition of percentage) I remember that 1% is actually 1/100th of something which I can write down using proper fraction as \(\frac{1}{100}\) or a decimal as 0.01.

Therefore any probability value from 0% to 100% can be written in these few forms. For instance:

• 0% = \(\frac{0}{100}\) = 0.00 = 0
• 1% = \(\frac{1}{100}\) = 0.01
• 5% = \(\frac{5}{100}\) = 0.05
• 10% = \(\frac{10}{100}\) = 0.10 = 0.1
• 20% = \(\frac{20}{100}\) = 0.20 = 0.2
• 50% = \(\frac{50}{100}\) = 0.50 = 0.5
• 100% = \(\frac{100}{100}\) = 1.00 = 1

To give you a better intuitive grasp of probability written as a decimal take a look at this simplistic graphical depiction of it

# prob = 0.0
impossible ||||||||||||||||||||||||||||||||||||||||||||||||||| certain
           ∆
# prob = 0.2
impossible ||||||||||||||||||||||||||||||||||||||||||||||||||| certain
                     ∆
# prob = 0.5
impossible ||||||||||||||||||||||||||||||||||||||||||||||||||| certain
                                    ∆
# prob = 0.8
impossible ||||||||||||||||||||||||||||||||||||||||||||||||||| certain
                                                   ∆
# prob = 1.0
impossible ||||||||||||||||||||||||||||||||||||||||||||||||||| certain
                                                             ∆

Anyway, when written down as a decimal (like a statistician would do it) the probability is easier to type with a keyboard and a software calculator. Additionally, now we will be able to perform some simple but useful calculations with those numbers (see the upcoming sections).