Probability Theory

Experiments and events

In physics, we encounter many laws having a deterministic character, as for example the law of falling bodies. This law allows us to very accurately predict the trajectory of a cannon bullet or a rocket. We consider that if we repeat the "experience", we will obtain exactly the same result.

On the contrary, in thermodynamics most of the laws are not of deterministic type. The thermodynamic phenomena are random; you cannot predict a result of such a phenomenon or process, but you may predict a distribution (la répartition) of results.

The biological laws (even the Mendelian inheritance) have all a random character.

By experiment we will understand a repeatable process, having an identifiable or a measurable result. Typical examples are:

– after throwing a dice we obtain an identifiable result;

– after measuring (in microns) the diameter of a cell, or the weight of a person(in grams), or the systolic pressure of a man, we obtain measurable results;

– after looking at a pulmonary radiography of a patient we obtain an identifiable result;

– after counting the flu cases detected by a physician we obtain an identifiable result (even if this is expressed as a number).

(Remember that measuring an object means comparing it with another object, previously chosen as a measure unit, and not counting!)

In most cases – and usually in biology – the result of an experiment is not unique and cannot be accurately predicted; we say that we have a random experiment (un expériment aléatoire). This means in fact that if we repeat several times a random experiment, each time we could obtain a different result.

By elementary event, we will understand a possible outcome (result) of a random experiment. An event is simply a collection of elementary events.

Obvious examples are obtained when throwing a dice: many ordinary people considered the appearance of as an "event". But, according to our definition above, the appearance of is an elementary event and, attention, an event is also the appearance of or ! After such an experiment, many events – not only six, but also thirty-six – could appear!

As another example, an "event" is the detection of a tumor when regarding the pulmonary radiography of our patient "John Johnson". (Here we have a simpler situation, only two outcomes are possible: we detect, or we do not detect the tumor. However, we will see that we have here, in theory, four different events!)

Consider for example the experiment consisting in measuring the diameter of a cell, which gives us a measurable result. If we choose an arbitrary interval [a, b] of real numbers – where a < b – we obtain the following event: the result of measuring a cell's diameter falls into our interval, that means between a and b. Denote by E this particular event.

From a logical point of view, we could consider the complementary event: the result of measuring the cell's diameter falls outside our interval, that means before a or after b. This complementary event of E will be denoted (read "E bar").

Of course, if our interval is [0, 1 (km)], then it is "certain" that the result of measuring cell's diameter will fall into this interval. In this case, we have the certain event.

The complementary of the certain event is called the impossible event and is denoted by the symbol  (the same used in set theory to denote the void set).

(When throwing a dice, the certain event consists in the appearance of any face; the impossible event consists in not appearing any face at all.)

Let us continue now to measure the diameter of cells and let us choose another interval [c, d] of real numbers (c < d). Denote by F the following event: the result of measuring a diameter falls in this latter interval, i.e. between c and d.

Now we have two genuine events, E and F (and automatically two complementary events, and ). But logic tells us that another event appears: the result of measuring a diameter falls between c and b (see figure attached).

It is natural to denote this latter event by and to say that it is the conjunction of events E and F. We will read this as "the event E and F".

In general, given the events E and F as outcomes of the same experiment, we can imagine another event as an outcome of our experiment.

For example, if we throw a dice, and if E denotes the appearance of a "less than three points" face, and F denotes the appearance of an "even" face, then is exactly the appearance of .

But, if E is the appearance of and F is the appearance of , then obviously is impossible, i.e.

In general, if for two events E and F we have

we say that our events are exclusive.