Sum of probabilities of outcomes = 1
Theory
The probabilities of all the possible outcomes of an experiment always add up to 1, because something is certain to happen. A missing probability is found by subtracting from \(1\): \(P(\text{not }A)=1-P(A)\).
When an experiment is carried out, one of the outcomes must occur, so together they are certain.
A certain event has probability \(1\), so the separate probabilities always total \(1\) (or \(100\%\) as percentages).
The complement of an event is “not that event”. Its probability is \(1-P(A)\), which is handy when only one outcome is left to find.
The complement is found by taking the probability away from the whole.
| \(P(A)\) | \(P(\text{not }A)\) |
|---|---|
| \(\dfrac{3}{4}\) | \(\dfrac{1}{4}\) |
| \(70\%\) | \(30\%\) |
| \(0.45\) | \(0.55\) |
How to find a missing probability
- Add the probabilities you already know.
- Subtract that total from \(1\) (or from \(100\%\)) to find what is left.
- Match the form first: write \(1\) as \(\dfrac{4}{4}\) before subtracting quarters.
| \(P(\text{lose})\) | \(=\) | \(1-\dfrac{3}{4}\) |
| \(=\) | \(\dfrac{4}{4}-\dfrac{3}{4}=\dfrac{1}{4}\) |
Rain and no rain are the only outcomes.
| \(100\%-70\%\) | \(=\) | \(30\%\) |
| \(0.45+0.25\) | \(=\) | \(0.7\) |
| \(P(\text{yellow})\) | \(=\) | \(1-0.7=0.3\) |
| \(30\%+45\%\) | \(=\) | \(75\%\) |
| \(100\%-75\%\) | \(=\) | \(25\%\) |
Common pitfalls
Frequently asked questions
Why do all the probabilities add to 1?
Because one of the outcomes must happen. That is certain, and certain has probability \(1\), so the parts total \(1\).
What is the complement of an event?
The complement is “not that event”. Its probability is \(1-P(A)\). If \(P(\text{rain})=0.7\), then \(P(\text{no rain})=0.3\).
How do I subtract a fraction from 1?
Write \(1\) with the same denominator first. For \(1-\dfrac{3}{4}\), use \(\dfrac{4}{4}-\dfrac{3}{4}=\dfrac{1}{4}\).
Do percentages also add to 100%?
Yes. As percentages the probabilities of all outcomes total \(100\%\), which is the same as \(1\).