What is probability?

Probability is simply how likely something is to happen. The probability of an event can range from being impossible, all the way to being certain.

We can express probabilities on a scale like the one above. Probabilities can be expressed as a decimal (0.25), a fraction (1/4), a percentage (25%), or in words ('unlikely').

Probability Notation

To express the probability of something happening mathematically, we write P(something). 

You will often see the something represented with an X, so you might see 'the probability of X as P(X)'.

For example, if the probability that it will rain today is 80%, or 0.8, then we would write: P(Rain) = 0.8.

Probabilities add up to 1

If you consider all possible outcomes of an event, then all the probabilities must add up to 1.

For example, if you flip a coin, the probability of it landing on heads and the probability of it landing on tails must add up to 1, because heads or tails are the only possible outcomes when you flip a coin.

You may sometimes see this idea expressed as:

P(event happens) + P(event doesn't happen) = 1

(be aware this only holds true if only one outcome can occur at a time. E.g. flipping a coin, which can only be heads or tails, it can't be both at once - or rolling a dice, which can only land on one number at a time)

How to calculate probabilities

To calculate the probability when all possible outcomes are equally likely, we can use the following formula:

Example: What is the probability of rolling a 3 on a fair six-sided die?

• There is only one side with '3' on the die, so the 'number of ways an outcome can happen' is one.
• There are six possible sides on the die. So the 'total number of possible outcomes' is six.

So our formula becomes:

So the chance of rolling a 3 is 1/6. We could also express this as 16.7%, or 0.167.

(you could also say it is 'unlikely', but in maths you should express probabilities as numbers whenever possible)