Probability as a fraction
Probability is a way of putting a number on how likely an event is. When every outcome is equally likely, you find it by counting:
The top is how many outcomes count as your event; the bottom is how many there are altogether. A fair die has \(6\) faces, and \(3\) of them are even, so:
- Event
- A collection of one or more outcomes we are interested in, such as "roll an even number" or "draw a red card". We write \(P(A)\) for the probability of event \(A\).
You can leave a probability as a fraction, or write the same value as a decimal or a percentage. A half is \(\tfrac{1}{2} = 0.5 = 50\%\). In an exam a tidy simplified fraction is usually the safest form.
The 0 to 1 scale
Every probability lands somewhere between \(0\) and \(1\) inclusive. Nothing can be more certain than certain, and nothing can be less likely than impossible.
- \(P = 0\) means the event is impossible. Rolling a \(7\) on an ordinary die.
- \(P = \tfrac{1}{2}\) means it is as likely as not. Flipping heads on a fair coin.
- \(P = 1\) means the event is certain. Rolling a number less than \(7\) on an ordinary die.
A second useful fact: the probabilities of all the possible outcomes of an experiment always add up to \(1\), because between them they cover everything that can happen.
A probability is never bigger than 1
If your answer comes out above \(1\) or below \(0\), something has gone wrong, usually the favourable count is larger than the total, or you have added when you should have multiplied. Stop and recount.
The complement: P(not A)
The complement of an event is everything in the sample space that is not that event. Since an event either happens or it does not, those two probabilities must add to \(1\). Rearranging gives a shortcut that saves a lot of counting:
This is priceless when the "not" case is easier to count than the case itself. Working out the chance of getting at least one of something is a classic example: it is far quicker to find the chance of getting none and subtract from \(1\).
A bag has 4 red, 5 blue and 3 green counters. One is drawn at random. Find the probability it is not blue.
Putting it together
Most simple-probability questions are one of three moves: count and divide, place the value on the \(0\) to \(1\) scale, or take a complement. The table below is a quick reference.
| You are asked for | Do this |
|---|---|
| The chance of an event | Favourable \(\div\) total, then simplify |
| How likely in words | Compare with \(0\), \(\tfrac{1}{2}\) and \(1\) |
| The chance it does not happen | \(1 - P(A)\) |
| The chance of "at least one" | \(1 - P(\text{none})\) |
Where this is assessed
Calculating a probability correctly and simplifying it is Criterion A (Knowing and understanding). Explaining your reasoning and setting out the complement clearly is Criterion C (Communicating). Deciding whether a real situation, like a raffle, is a fair deal draws on Criterion D (Applying in real-life contexts).
Check yourself
Work each out before revealing the answer.
1. A fair die is rolled. What is the probability of a number greater than 4? +
Numbers greater than \(4\) are \(5\) and \(6\), so \(2\) favourable outcomes out of \(6\). \(P = \dfrac{2}{6} = \) \(\dfrac{1}{3}\).
2. The probability it rains tomorrow is 0.35. What is the probability it does not rain? +
Use the complement rule: \(P(\text{no rain}) = 1 - 0.35 = \) \(0.65\).
3. A bag has 7 white and 3 black beads. A bead is drawn at random. Find the probability it is not white. +
Total \(= 7 + 3 = 10\). \(P(\text{white}) = \dfrac{7}{10}\), so \(P(\text{not white}) = 1 - \dfrac{7}{10} = \) \(\dfrac{3}{10}\). This matches counting the \(3\) black beads directly.
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.