## Introduction

The history of probability theory began with card, dice and roulette games. This is why there are so many examples of gambling in the study of probability. Probability theory allows one to calculate the chance of a number occurring in a randomized experiment.

## Random experiment

It is that experiment that, when repeated under the same conditions, can yield different results, ie, results explained at random. When it comes to time and chances of winning the lottery, the approach involves random experiment calculation.

## Sample space

It is the set of all possible results of a randomized experiment. The letter representing the sample space is S.

**Example:**

Rolling a coin and a dice simultaneously, where S is the sample space, consisting of the 12 elements:

S = {K1, K2, K3, K4, K5, K6, R1, R2, R3, R4, R5, R6}

- Explicitly write the following events:

A = {guys and an even number appears}

B = {a prime number appears}

C = {crowns and odd number appear} - Idem, the event in which:

a) A or B occur;

b) B and C occur;

c) Only B occurs.

- Which of events A, B and C are mutually exclusive?

**Resolution:**

- To obtain A, we choose the elements of S consisting of a K and an even number: A = {K2, K4, K6};

To obtain B, we choose the points of S consisting of prime numbers: B = {K2, K3, K5, R2, R3, R5};

To obtain C, we choose the points of S consisting of an R and an odd number: C = {R1, R3, R5}.

- (a) A or B = AUB = {K2, K4, K6, K3, K5, R2, R3, R5}

(b) B and C = BC = {R3, R5}

(c) We choose the elements of B that are not in A or C:

B THE^{ç }Ç^{ç }= {K3, K5, R2}

- A and C are mutually exclusive because A C =

## Probability Concept

If in a random phenomenon the possibilities are equally likely, then the probability of an event occurring is:

For example, when rolling a die, an even number may occur in 3 different ways out of 6 equally likely, so P = 3/6 = 1/2 = 50%.

We say that a sample space S (finite) is equiprobable when its elementary events have equal probability of occurrence. In an equiprobable sample space S (finite), the probability of occurrence of an event A is always:

### Important properties:

1. If A and A 'are complementary events, then:

*P (A) + P (A ') = 1*

2. The probability of an event is always a number between 0 (impossible event probability) and 1 (right event probability).

Next: Conditional Probability