Probability

Probability is a way of measuring how likely something is to happen. Every probability is a number on a scale from 0 to 1.

• A probability of 0 means the event is impossible — it cannot happen.
• A probability of 1 means the event is certain — it will definitely happen.
• A probability of $\tfrac{1}{2}$ (or 0.5) means an even chance — just as likely to happen as not.

A probability can never be less than 0 or more than 1. If you ever calculate a probability of $1.5$ or $-0.2$, something has gone wrong.

You can write a probability as a fraction, a decimal or a percentage — they all mean the same thing. An even chance is $\tfrac{1}{2} = 0.5 = 50\%$.

Before you put numbers to probability, it helps to describe chance in words. These words match positions on the probability scale:

• Impossible — it cannot happen (probability 0).
• Unlikely — it probably will not happen (probability between 0 and $\tfrac{1}{2}$).
• Even chance — equally likely to happen or not (probability $\tfrac{1}{2}$).
• Likely — it probably will happen (probability between $\tfrac{1}{2}$ and 1).
• Certain — it will definitely happen (probability 1).

Think about some everyday events:

• The sun rising tomorrow — certain.
• Rolling a 7 on an ordinary dice — impossible (a dice only has 1 to 6).
• A flipped coin landing on heads — even chance.
• Picking a red sweet from a bag with mostly red sweets — likely.

The word fair is important too. A fair coin or dice gives every outcome an equal chance. A biased one does not — perhaps it is weighted to favour a certain result.

Using the right word shows you understand roughly how likely an event is — and that is the first step before working out the exact number.

To calculate probability, you need two clear ideas.

An outcome is a single possible result of an experiment. When you roll an ordinary dice, the possible outcomes are 1, 2, 3, 4, 5 and 6 — six outcomes in total.

An event is the thing you are interested in. It can be a single outcome, or a group of outcomes. "Rolling a 4" is an event made of one outcome. "Rolling an even number" is an event made of three outcomes — 2, 4 and 6.

Outcomes are equally likely when each has the same chance — every face of a fair dice, for example. This is the situation you need before you can use the probability calculation in the next section.

When all outcomes are equally likely, there is a simple formula for the probability of an event:

$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$

A favourable outcome is one that counts as the event happening.

Roll a fair dice. What is the probability of rolling an even number?

• Favourable outcomes (even numbers): 2, 4, 6 — that is 3.
• Total outcomes: 1, 2, 3, 4, 5, 6 — that is 6.
• So $P(\text{even}) = \tfrac{3}{6} = \tfrac{1}{2}$.

For the bag above — 3 red counters and 7 blue, 10 in total — the probability of picking a red counter is $P(\text{red}) = \tfrac{3}{10} = 0.3 = 30\%$.

A useful check: the probabilities of all possible outcomes always add up to 1. Here $\tfrac{3}{10} + \tfrac{7}{10} = 1$.

To find the total number of outcomes, you often need to list them all. The secret is to be systematic — work in a fixed order so nothing is missed and nothing is counted twice.

Suppose you flip a coin and roll a dice. List the outcomes by taking each coin result in turn and pairing it with every dice number:

There are 12 outcomes in total. A table like this — or an ordered list — guarantees you have found them all.

Once the full list is written, calculating a probability is easy. The probability of getting heads and an even number is $\tfrac{3}{12} = \tfrac{1}{4}$, because three outcomes (H,2 / H,4 / H,6) are favourable out of 12.

Understanding probability now sets you up for plenty of maths ahead — and for thinking clearly about the world:

• Data handling — probability and statistics work together; data you collect can help you estimate how likely events are.
• Fractions, decimals and percentages — every probability is written in one of these forms, so this topic gives them a real purpose.
• Later probability work — you will go on to combine events and compare expected results with what actually happens.
• In everyday life, probability shapes weather forecasts, games, sports predictions and sensible decision-making.

Whenever you hear "there is a 70% chance of rain" or "you are unlikely to win", you now know exactly what those statements mean — and how someone worked them out.