2.1.3.2.4 - Complements

Complement

The probability that the event does not occur. The complement of \(P(A)\) is \(P(A^C)\). This may also be written as \(P(A')\).

In the diagram below we can see that \(A^{C}\) is everything in the sample space that is not A.

Complement of A
Complement of A
\(P(A^{C})=1−P(A)\)

Example: Coin Flip Section

When flipping a coin, one can flip heads or tails. Thus, \(P(Tails^{C})=P(Heads)\) and \(P(Heads^{C})=P(Tails)\)

Example: Hearts Section

If you randomly select a card from a standard 52-card deck, you could pull a heart, diamond, spade, or club. The complement of pulling a heart is the probability of pulling a diamond, spade, or club. In other words: \(P(Heart^{C})=P(Diamond,\; Spade,\;\;Club)\)

The complement of any outcome is equal to one minus the outcome. In other words: \(P(A^{C})=1-P(A)\)

It is also true then that: \(P(A)=1-P(A^{C})\)

Example: Rain Section

Light Rain Showers

According to the weather report, there is a 30% chance of rain today: \(P(Rain) = .30\) 

Raining and not raining are complements.

\(P(Not \:rain)=P(Rain^{C})=1-P(Rain)=1-.30=.70\)

There is a 70% chance that it will not rain today.

Example: Winning Section

The probability that your team will win their next game is calculated to be .45, in other words:

\(P(Winning)=.45\)

Winning and losing are complements of one another. Therefore the probability that they will lose is:

\(P(Losing)=P(Winning^{C})=1-.45=.55\)

The sum of all of the probabilities for possible events is equal to 1.

Example: Cards Section

In a standard 52-card deck there are 26 black cards and 26 red cards. All cards are either black or red.

\(P(red)+P(black)=\frac{26}{52}+\frac{26}{52}=1\)

Example: Dominant Hand Section

Of individuals with two hands, it is possible to be right-handed, left-handed, or ambidextrous. Assuming that these are the only three possibilities and that there is no overlap between any of these possibilities:

\(P(right\;handed)+P(left\;handed)+P(ambidextrous) = 1\)