# Probability Theory

### Deterministic Experiments

With deterministic experiments, the results can be predicted before the experiment is conducted.

#### Example

If a stone is dropped from a window, it is know, undoubtedly, that the stone will go down. If it is throw up in the air, it is know that it will travel upwards over a certain time interval; but afterwards will come down.

### Random Experiment

With random experiments, the results cannot be predicted, since they depend on chance.

#### Examples

If a coin is flipped, it is not known beforehand whether it will be heads or tails.

Similarly, if a die is rolled, the result cannot be determined beforehand.

### Probability Theory

Probability theory deals with assigning a number to each possible result that can occur in a random experiment. With this being said, the following definitions need to be introduced:

#### Outcome

An outcome is each of the possible results of a random experiment.

Obtaining heads when flipping a coin.

Obtaining a 4 when rolling a die.

#### Sample Space

The sample space is the set of all possible outcomes of a random experiment. It is denoted by S (or by the Greek letter Ω).

Sample space of a coin:

S = {H, T}.

Sample space of a die:

S = {1, 2, 3, 4, 5, 6}.

#### Event

An event is any subset of the sample space.

For example, when rolling a die, an event would be the outcome of an even number, or another, obtaining a multiple of 3.

#### Example

A bag contains blue and red balls. Three balls are drawn successively. Calculate:

1. The sample space.

S = {(b,b,b); (b,b,r); (b,r,b); (r,b,b); (b,r,r); (r,b,r); (r,r ,b); (r, r,r)}

2. The event A = (draw three balls of the same color).

A = {(b,b,b); (r, r,r)}

3. The event B = (extract at least one blue ball).

B= {(b,b,b); (b,b,r); (b,r,b); (r,b,b); (b,r,r); (r,b,r); (r,r ,b)}

4. The event C = {extract only one red}.

C = {(b,b,r); (b,r,b); (r,b,b)}