Event Probability

Each event E in a sample space S is associated with a probability, denoted as P(E).

When an experiment is repeated a large number of times, the event tends to occur with a stable relative frequency, which corresponds to its probability P(E).

The probability of an event must adhere to certain axioms:

Axiom 1
The probability of an event is a value between 0 and 1. $$ 0 \le P(E) \le 1 $$
Axiom 2
The total probability of all events in a sample space is 1. $$ P(S)=1 $$
Axiom 3
If the sample space consists of a finite set of mutually exclusive events, meaning if one event E1 occurs, the others cannot, then the probability that at least one of these events occurs is the sum of their individual probabilities. $$ P( U_{i=1}^n E_i ) = \sum_{i=1}^n P(E_i) $$
Example: When rolling a die, event E occurs if a 2 is rolled, while event F occurs if a 1 is rolled. The probabilities of these events are P(E)=1/6 and P(F)=1/6. Since the two events are mutually exclusive, the probability of rolling either a 1 or a 2 is P(E)+P(F)=1/6+1/6=1/3.

Key Insights

Here are a few important observations:

When two events are not mutually exclusive, they can occur simultaneously.
In this situation, to determine the probability of either event occurring, adjust the formula: $$ P(E∪F) = P(E)+P(F)-P(E⋂F) $$
To find the probability that an event E does not occur, you need to consider its complement, E^c. The complement E^c includes all outcomes in the sample space except E. Thus, the probability that event E does not happen is 1 minus the probability of E. $$ P(E^c) = 1 - P(E) $$
Example: If the sample space for rolling a die includes six outcomes, one for each face, and event E represents rolling a 2, with a probability of P(E)=1/6, then the probability that a 2 is not rolled is P(E^c)=1-1/6=5/6.

Example

35% of the population owns a car (event E), 10% owns a motorcycle (event F), and 5% own both a car and a motorcycle (E ⋂ F).

What percentage of the population owns neither a car nor a motorcycle?

First, calculate the percentage of people who own either a car or a motorcycle.

$$ P(E ∪ F) = P(E)+P(F)-P(E ⋂ F) = 0.35+0.10-0.05 = 0.40 $$

So, 40% of the population owns at least one car or motorcycle.

As a result, 60% of the population owns neither a car nor a motorcycle.

$$ 1 - 0.4 = 0.6 $$

And so forth.