This** math course on probability** is available for free download in pdf format.

## I. Calculating frequencies

In a **bivariate statistical series**, the values are usually represented in a cross tabulation.

The sums of the rows and columns of a double entry table are called the table **margins**.

They appear in yellow in the table below.

The **marginal frequency** of a value is the quotient of the total number of this value by the total number of people.

Remark:

We speak of marginal frequency because we use only the numbers located in the margin of the table.

Example:

We consider a first year class of 32 students having chosen or not the HGGSP speciality.

Of the 32 students in the class, 21 chose the HGGSP specialty.

The marginal frequency of the value “HGGSP specialty” is therefore .

Of the 32 students in the class, 14 are girls.

The marginal frequency of the value “girls” is therefore equal to or 43.75%.

When looking for the frequency of occurrence of the value A only for a non

If the value of A is the same as the value of B in the statistical series, we say that we calculate the **conditional frequency of the value A among B**.

This conditional frequency, noted , is equal to .

Remark:

We speak of **conditional frequency** because we calculate the frequency of a value by imposing a condition.

Example:

We take the example above and we try to find out the frequency of girls (value A) among the students

who did not choose the HGGSP specialty (sub-population B).

In the table, we read that there are 5 girls who did not choose the HGGSP specialty out of a total of 11 students who do not follow this specialty.

5

Thus, .

Of the students who are not enrolled in HGGSP, approximately 45.5% are female.

## II. Calculating probabilities

Let A and B be two events in the same universe with non-zero probability.

The **conditional probability** that event B will occur knowing that event A has already occurred is noted and is defined by .

Example:

We take the previous example. We choose a student from the class at random and consider the events:

A: “The student has chosen the HGGSP specialty” and B: “The student is a boy”.

We use the table to find .

The probability of choosing a boy knowing that the chosen student is taking the HGGSP specialty is .

When performing a random experiment involving several events, it is easier to organize the different outcomes using a **probability tree**.

The first set of branches separates the outcomes according to the realization of the first event.

The second set of branches according to the second event, etc.

We indicate on each branch of the tree the corresponding probability as shown on the tree below.

The probabilities of the second level of the tree are **conditional probabilities**.

1. In a probability tree, the sum of the probabilities on the branches coming from the same node is

equal to 1.

2. A **path** is a sequence of branches describing a succession of events. The probability of a

path is equal to the product of the probabilities located on the branches that compose it.

The probability of an event is the sum of the probabilities of the paths that lead to it.

Events A and B are said to be **independent** when or, symmetrically, when .

Remark:

Intuitively, this means that the probability that B will happen does not depend on the realization of event A.

Example:

Keeping the same example, we observe that and .

We deduce that the events A and B are not independent.

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