Basic Probability

The probability of an event is determined by dividing the count of favorable outcomes by the count of all possible outcomes, assuming each outcome has an equal chance of occurring.

basic probability

Given that all outcomes are equally likely, we can compute the probability of an event [latex]E[/latex] using this formula:

 

[latex]P(E)=\frac{\text{Number of outcomes corresponding to the event E}}{\text{Total number of equally-likely outcomes}}[/latex]

 

Probabilities can be expressed as decimals, fractions, or percentages.

Notation: The probability of an event is notated as [latex]P(\text{event})[/latex]

Adding and subtracting fractions with common denominators: [latex]\dfrac{a}{c}\pm \dfrac{b}{c}=\dfrac{a\pm b}{c}[/latex]

Note that this relationship is described in both directions in the two equations below. That is, it is also true that [latex]\dfrac{a\pm b}{c}=\dfrac{a}{c}\pm \dfrac{b}{c}[/latex]. The second equation furthermore includes the fact that [latex]\dfrac{a}{a}=1[/latex].

Probability, likelihood, and chance are related concepts that are often used interchangeably, but they have distinct meanings. Probability is a mathematical concept used to quantify the likelihood of an event occurring, likelihood is a measure of how well a hypothesis or model fits the data, and chance is an informal way of expressing the uncertainty of an event. For this lesson, we will be focusing on probability.

In the realm of probability, certain and impossible events represent the extremes of what can occur. An event that cannot happen is deemed impossible, hence it has a probability of [latex]0[/latex], while an event that is sure to happen is certain, with a probability of [latex]1[/latex]. All other events fall somewhere between these two extremes, with their probability values reflecting how likely they are to occur.

certain and impossible events

• An impossible event has a probability of [latex]0[/latex].
• A certain event has a probability of [latex]1[/latex].
• The probability of any event must be [latex]0\le P(E)\le 1[/latex]

Complement of an Event

The concept of the complement of an event in probability is crucial as it helps to understand the likelihood of an event not occurring. By defining the complement, denoted by [latex]\bar{E}[/latex] , we can easily calculate its probability by subtracting the probability of the event [latex]E[/latex] from one. This relationship is fundamental in probability theory, as it connects the occurrence and non-occurrence of an event, completing the picture of all possible outcomes.

complementary events

The complement of an event is the event “[latex]E[/latex] doesn’t happen”

 

• The notation [latex]\bar{E}[/latex] is used for the complement of event [latex]E[/latex].
• We can compute the probability of the complement using [latex]P\left({\bar{E}}\right)=1-P(E)[/latex]
• Notice also that [latex]P(E)=1-P\left({\bar{E}}\right)[/latex]