Probability

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
	2

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
	6

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
	14

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
A^C

Event	Complement
Win	Lose
Rain	No Rain
Flip heads on a coin	Flip 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)