Probability: part 1

To quantify results, we may use a variety of methods, terms, and notations, but a few common ones are:

• a percentage (for example '50%')
• a proportion of the total number of outcomes (for example, '5/10')
• a proportion of 1 (for example, '1/2')

You may notice that all three of the above examples represent the same probability, and in fact ANY method of probability isfundamentally based on the following procedure:

1. Define a process.
2. Define the total measure for all outcomes of the process.
3. Describe the likelihood of each possible outcome of the process with respect to the total measure.

The term 'measure' may be confusing, but one may think of it as a ruler. If we take a ruler that is 1 metre long,then half of that ruler is 50 centimetres, a quarter of that ruler is 25 centimetres, etc. However, the thing to remember is that without theruler, it makes no sense to talk about proportions of a ruler! Indeed, the three examples above (50%, 5/10, and 1/2) represented the same probability,the only difference was how the total measure (ruler) was defined. If we go back to thinking about it in terms of a ruler '50%' means '50/100', so it meanswe are using 50 parts of the original 100 parts (centimetres) to quantify the outcome in question. '5/10' means 5 parts out of the original 10 parts (10centimetre pieces) depict the outcome in question. And in the last example, '1/2' means we are dividing the ruler into two pieces and saying that oneof those two pieces represents the outcome in question. But these are all simply different ways to talk about the same 50 centimetres of the original 100centimetres! In terms of probability theory, we are only interested in proportions of a whole .

Although there are many ways to define a 'measure', the most common and easiest one to generalize is to use '1' as the total measure. So if weconsider the coin-flip, we would say that (assuming the coin was fair) thelikelihood of heads is 1/2 (i.e. half of one) and the likelihood of tails is 1/2. On the other hand, if we consider the event of not flipping the coin, then(assuming the coin was originally heads-side-up) the likelihood of heads is now 1, while the likelihood of tails is 0. But we could have also used '14' as theoriginal measure and said that the likelihood of heads or tails on the coin-flip was each '7 out of 14', while on the non-coin-flip the likelihood of heads was'14 out of 14', and the likelihood of tails was '0 out of 14'. Similarly, if we consider the throwing of a (fair) six-sided die, it may be easiest to set thetotal measure to '6' and say that the likelihood of throwing a '4' is '1 out of the 6', but usually we simply say that it is 1/6, i.e. '1/6 of 1'.

Definitions

There are three important concepts associated with a random experiment: 'outcome', 'sample space', and 'event'. Two examples ofexperiments will be used to familiarize you with these terms:

• In Experiment 1 a single die is thrown and the value of the top face after it has come to restis noted.
• In Experiment 2 two dice are thrown at the same time and the total of the values of each of the top faces after they have come torest is noted.