Activities

Conditional probability - notation, multiplication law, dependent and independent events

Consider a company with 60 employees (45 males and 15 females).

There are three types of employee:
  • A full time - 10 employees
  • B part time - 20 employees
  • C casual - 30 employees
A
B
C
M
F
7
15
23
3
5
7

Suppose an employee is chosen at random.

Consider events A = {employee is full time}, M = {employee is male}, etc.

Let the sample space of all employees be represented by S.
 

If the employee chosen is full time, what is the probability that the employee is a female?

As event A has occurred, the sample space is restricted to the 10 outcomes of the event {employee is full time}. Of these outcomes, there are 3 in the event {employee is female and full time}.

A vertical line is used to represent "if". The event "F and A" is the intersection or overlap of the two events and the  symbol is used.
 

This result can be rearranged.

The working opposite demonstrates the multiplication law.

There is a second version.

Consider two separate questions:

The questions have the same answer because events F and B are independent.

Knowing that the employee is part time does not effect the probability of the employee being female. This is because the proportion of part time employees that are female is the same as the proportion for the whole company i.e. one in four or 1/4.

Two events are independent if the occurrence of one has no effect on the probability of the other.

The multiplication law  has a special form for independent events.
 

Multiplication law for independent events