The Probability Scale

1 · How likely is it? Words for chance

You already talk about chance all the time. "It'll probably rain." "There's no way we'll lose." "It's definitely Friday tomorrow." Each of these is a guess about how likely something is to happen.

In maths, the thing we are interested in (like "it rains tomorrow" or "I roll a 6") is called an event. The probability of an event is a measure of how likely it is to happen. Before we use numbers, here are the five words mathematicians use, from least likely to most likely:

• Impossible: it cannot happen at all. (Rolling a 9 on an ordinary dice.)
• Unlikely: it could happen, but it happens less often than not. (Rolling a 6.)
• Even chance: it is just as likely to happen as not to happen. People often say "50–50". (A coin landing on heads.)
• Likely: it happens more often than not, but it is not guaranteed. (Rolling a number bigger than 1.)
• Certain: it will definitely happen, every single time. (Rolling a number less than 7.)

Try it: read each event and pick the word that fits best. The line underneath shows where each word lives, from "can never happen" on the left to "always happens" on the right.

2 · The probability scale from 0 to 1

Mathematicians give every probability a number between 0 and 1. Picture a line, the probability scale:

• An impossible event has probability 0. It sits at the left-hand end.
• A certain event has probability 1. It sits at the right-hand end.
• An even chance event has probability 0.5, which is the same as 1/2. It sits exactly halfway.
• Unlikely events are somewhere between 0 and 0.5; likely events are somewhere between 0.5 and 1.

The further right, the more likely. An event at 0.9 is very likely; an event at 0.1 is very unlikely. Drag the teal marker (or use the slider) to where you think each event belongs, then press Check. Correct events stay on the scale as numbered plum dots.

Some of those you could work out exactly (dice, coins, cards and spinners), and the next two sections show you how. Others, like the weather or a lottery, can only be estimated from what has happened before. That is a different kind of probability, which you can explore on the theoretical and experimental probability page.

3 · Outcomes, events and equally likely

To work out a probability exactly we need a few more words. Think about rolling one ordinary dice once.

• An outcome is one possible result. Rolling a dice has 6 outcomes: 1, 2, 3, 4, 5 and 6.
• An event is one or more outcomes grouped together. "Rolling an even number" is an event made of the outcomes 2, 4 and 6.
• The favourable outcomes are the outcomes that make the event happen. For "rolling an even number" the favourable outcomes are 2, 4 and 6. ("Favourable" doesn't mean good; it just means "the ones we're counting".)
• Outcomes are equally likely if each one has exactly the same chance. On a fair dice each number is equally likely. A dice that has been weighted so that 6 comes up more often is called biased.

Choose an event, or tap the dice yourself to make up your own event. The orange faces are the favourable outcomes; all 6 faces together are the total.

4 · Calculating a probability

Imagine a bag with 3 orange counters and 1 purple counter. You shake it and pick one counter without looking. We say you pick at random: every counter has the same chance of being picked.

There are 4 counters, so 4 equally likely outcomes. 3 of them are orange. So, on average, 3 picks out of every 4 will be orange. We write this as

P(orange) = 3/4

and read it as "the probability of picking orange is three quarters". The P( ) notation is short for "the probability of". The rule behind what we just did is:

So the method is always the same four steps:

1. Count the total number of equally likely outcomes.
2. Count the favourable outcomes (the ones where the event happens).
3. Write favourable ÷ total as a fraction.
4. Simplify the fraction if you can, and change it to a decimal or percentage if you're asked to.

Build your own bag below. The spinner next to it has one equal section for each counter, so it gives exactly the same probabilities. Watch the teal marker move along the scale as you change the numbers.

Things to try: make the bag all one colour (the probability jumps to 1, certain). Set a colour to 0 (its probability is 0, impossible). Make two colours equal and the third 0 (even chance). Notice too that if you add up P(orange), P(purple) and P(teal) you always get exactly 1. Why that happens, and how to use it, is on the mutually exclusive outcomes page.

5 · Fractions, decimals and percentages

A probability is a number between 0 and 1, so it can be written in any of three ways, and they all mean the same point on the scale:

• as a fraction: 1/4
• as a decimal: 1 ÷ 4 = 0.25
• as a percentage: 0.25 × 100 = 25%

If you need a reminder of how to convert between them, see fractions, decimals and percentages. Here are the ones you'll meet most (≈ means "approximately"; these two percentages are rounded to 1 d.p.):

Type any probability, as a fraction (3/8), a decimal (0.6) or a percentage (45%), and see it on the scale in all three forms.

Which is more likely?

To compare probabilities written in different forms, change them all to decimals (or all to percentages) first. Then the bigger number is the more likely event. Tap the more likely one.

6 · Nothing below 0, nothing above 1

Think about the fraction favourable ÷ total. The favourable outcomes are some of the total outcomes, so:

• the top can never be bigger than the bottom (you can't have 5 winning tickets out of 3 tickets), so the probability can never be more than 1;
• the top can't be negative (you can't have −2 ways of winning), so the probability can never be less than 0.

So answers like 3/2, 1.4, 120% or −0.3 are always wrong. Nothing is "more than certain" or "less than impossible". (Footballers who promise to "give 110%" are not doing probability.) If you get an answer like that, you have probably put the fraction upside down: check your working.

Could each of these be a probability? Decide, and see where it would land.

"There are two outcomes, so it's 50–50"

This is the most common probability mistake of all. "I'll either win the raffle or I won't, so my chance is 1/2." "Either it snows tomorrow or it doesn't, so it's even chance." Having two outcomes does not make them equally likely. The 50–50 idea only works when both outcomes have exactly the same chance, like heads and tails on a fair coin.

This spinner has 10 equal sections. Each section says WIN or LOSE, so there are only ever two results. Change how many sections are WIN and watch the real probability move away from the dashed "50–50?" guess.

7 · Calculate it: practice

Now you try. Each question is new. Count the total outcomes, count the favourable ones, then divide. You can answer as a fraction (2/5), a decimal (0.4) or a percentage (40%). If you go wrong, the feedback tries to spot which mistake you made.

Worked examples

Quick check

What you've learned

• The probability of an event measures how likely it is to happen.
• Words for chance, from least to most likely: impossible, unlikely, even chance, likely, certain.
• The probability scale runs from 0 (impossible) to 1 (certain), with even chance at 0.5. The closer to 1, the more likely.
• An outcome is one possible result; an event is a group of outcomes; the favourable outcomes are the ones that make the event happen.
• When all outcomes are equally likely: P(event) = number of favourable outcomes ÷ total number of outcomes. Always divide by the total, and simplify the fraction.
• A probability can be written as a fraction, a decimal or a percentage (1/4 = 0.25 = 25%), but not as a ratio. Convert to decimals to compare.
• A probability can never be less than 0 or more than 1 (or more than 100%). An answer like 3/2 means something has gone wrong.
• Two possible outcomes does not mean 50–50, unless the two outcomes are equally likely.