Probability Definition:

The likelihood of something happening or being the case.
Examples include probability of flipping a head, rolling a 6 on a single cube, or being born on a Sunday.

Probability Terms to Know:

Experiment: a repeatable process with a set of possible results
Outcome: A possible result of an experiment
Sample Space: all the possible outcomes of an experiment
Event: one or more outcomes of an experiment

General Probability Formula

Probability of an event happening  =  Number of ways the event can happen
  Total Number of Outcomes

How To Write Probabilities:

Probability values can be written as a decimal, fraction, or percentage.

Flip 1 Coin Example

A coin has 2 sides. 1 head, and 1 tail. So we have:
Probability of Heads  =  Total number of heads
  Total number of coin faces

Probability of Heads  =  1

This can also be written as 50% or 0.5

Roll Dice (Cube) Example:

A die/cube has 6 sides (1, 2, 3, 4, 5, 6) so we have:
Probability of 3  =  Total number of 3's
  Total number of die/cube faces

Probability of 3  =  1

This can also be written as 16.67% or 0.1667

Equally likely events:

For equally likely events, like coin flips and die rolls for instance, the probabilty for each event is 1/N where N is the number of possible outcomes

Fruit in a Bowl Example:

Suppose we have a bowl of fruit with 3 apples, 5 oranges, and 6 bananas
We want to find out the probability of picking an orange

Probability of picking an Orange  =  Total oranges
  Total fruits

Probability of picking an Orange  =  5 oranges
  3 apples + 5 oranges + 6 bananas

Probability of picking an Orange  =  5

This can also be written as 35.71% or 0.3571

Probability Event Postulate:

For an Event A, 0 ≤ P(A) ≤ 1
A probability of 0 means the event is impossible.
A probability of 1 means the event is certain.
A probability of 0.5 or 1/2 or 50% means the event is equally likely to happen as it is not happen.
A probability greater than 1/2 or 0.5 or 50% and less than 1 is likely to happen.
A probability less than 1/2 or 0.5 or 50% and greater than 0 is unlikely to happen.

Sample Space Postulate:

Sample Space: the set of all possible outcomes or results of that experiment.
For a Sample Space S (all possible outcomes), P(S) = 1 (since it is all possible outcomes)

Empty Set Postulate:

Empty Set: The set with no elements
Probability of the empty set (event without outcomes) is: P(∅) = 0

Complement of an event:

Complement of an event: The opposite of an event happening
RainNo Rain
Flip heads on a coinFlip tails on a coin

Probability of the complement:

Given an Event A, the complement, A', is anything in the sample space which is not A
P(A') = 1 - P(A)

Proof of the Probaility of the complement:

P(S) = 1 By the sample space postulate above
P(A U A') = 1
P(A) + P(A') = 1
P(A') = 1 - P(A)

How does the Probability Calculator work?

This lesson walks you through the basics of probability like the probability definition, events, outcomes, experiments, and probability postulates

What 5 formulas are used for the Probability Calculator?

  1. Probability of an event happening = Number of ways the event can happen/Total Number of Outcomes
  2. For an Event A, 0 ≤ P(A) ≤ 1
  3. P(S) = 1
  4. P(∅) = 0
  5. P(A') = 1 - P(A)

For more math formulas, check out our Formula Dossier

What 10 concepts are covered in the Probability Calculator?

The opposite of an event happening
empty set
The set with no elements
a set of outcomes of an experiment to which a probability is assigned.
In statistics, a controlled and repeatable process
how likely a particular population is to produce an observed sample
a possible result of an experiment or trial
A statement accepted as true without proof
the likelihood of an event happening. This value is always between 0 and 1.
P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomes
sample space
the set of all possible outcomes or results of that experiment.
Statistics is a discipline concerned with the analysis of data and decision making based upon data.

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