What is the complement of an event A and how do you denote it?

The complement of an event A is all outcomes that are not the event A. This is denoted by:

With a sample space S, then the complement of A are all of the outcomes not in event A.

For example, if S = {a, b, c, d, e} and A = {a, b, c}, then: It is usually helpful to use a Venn diagram to visualise this, where the complement of A is all outcomes outside of A (shaded grey):

Also, the probability of the complement of A is equal to 1 (which is the probability of an outcome in the sample space) minus the probability of A:

The complement of an event is the subset of outcomes in the sample space that are not in the event. A complement is itself an event.

For example-

If our sample space is rolling a regular six sided dice, Event A could be rolling a 1,2 or 5.

Thus, the complement of A would be the event of rolling a 3,4 or 6. This can be denoted as A’ or A^c.

The complement of an event A is “not A”, the complete opposite of A.

Eg. If event A = "my team win their next match", then the complement of A = "my team do not win their next match (they lose or draw)".

We write the complement of A as A'

The complement of A is the probability that A doesn't happen. It is donated by P(A').

P(A') = 1- P(A)

An example: if you have the numbers 1,2,3,4,5,6,7,8,9,10 as you event space. Then if A are the numbers that have 3 as a factor i.e. 3,6,9 then P(A) = 3/10=0.3

So the P(A') = 1- 0.3 = 0.7=7/10 because there are 7 numbers out of the the ten that don't divide by three namely 1,2,4,5,7,8,10

Complement of an event A is expressed as A'