Random experiments  (Page 2/3)

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 is fundamentally 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 metrestick, then half of that metrestick is 50 centimetres, a quarter of that metrestick is 25 centimetres, etc. However, the thing to remember is that without the metrestick, it makes no sense to talk about proportions of a metrestick! 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 metrestick '50%' means '50/100', so it means we 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 (10 centimetre pieces) depict the outcome in question. And in the last example, '1/2' means we are dividing the metrestick into two pieces and saying that one of 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 100 centimetres! 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 we consider the coin-flip, we would say that (assuming the coin was fair) the likelihood 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 the original 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 the total 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.

Random experiments

The term random experiment or statistical experiment is used to describe any repeatable process, the results of which are analyzed in some way. For example, flipping a coin and noting whether or not it lands heads-up is a random experiment because the process is repeatable. On the other hand, your reading this sentence for the first time and noting whether you understand it is not a random experiment because it is not repeatable (though making a number of random people read it and noting which ones understand it would turn it into a random experiment).