The maths behind everyday decisions.

A day of quiet maths

Do you use mathematics outside the classroom? Most people say no. Then think about yesterday. Did you guess whether it would rain before you left home? That was probability. Did you look at a chart, a score or a price and decide whether it was good? That was statistics. Did you think about what a friend would do before you made a move in a game? That was game theory. Did you unlock a phone? That was cryptography, working for you without a word.

So you already use mathematics, many times a day. You just do not call it that. This course is about that quiet mathematics: the kind that runs under everyday decisions, and the kind that people use to persuade you, sometimes honestly and sometimes not.

What does your gut say?

Three quick questions before we begin. There is no score, and nothing leaves this page. At the end of the course you will see your first answers next to what you know by then.

How this course works

Four parts and 26 chapters, then a wrap-up, about 12 hours, at your own pace. Each chapter has:

• a story, something to try and a question to argue about,
• an "In short" box at the top, so you can see the main points first,
• a link into World 101, Learnacy's map of what the world knows, and often a Learnacy Lab to play with,
• a note for your notebook, and
• a five-question check. Pass it with 4 right, as often as you need.

Pass every chapter check and the final round, and you get a certificate of completion with a link anyone can use to verify it.

Two promises. Every game of chance in this course uses made-up points, never money, and you will see why games of chance lose on average. And every worked example tells you whether it is made up or comes from a real source.

One rule for the whole course: questions are more important than answers. In mathematics, a good question ("how would I know?") is often worth more than a quick answer.

The unfinished game

Two friends are playing a game of chance for points. The first to win three rounds takes all 100 points. One friend has won two rounds and the other has won one when the bell rings, and they have to stop. How should they split the 100 points fairly?

Split it half and half? That is unfair to the one who is ahead. Give it all to the leader? That is unfair to the other, who could still have won. So what is fair?

This is a made-up version of a real puzzle. In 1654, two French mathematicians, Blaise Pascal in Paris and Pierre de Fermat in Toulouse, wrote letters to each other through the summer and autumn about exactly this kind of problem: how to split the money fairly when a game stops before it is finished. They used two different methods, and their answers agreed. What was new was the way they got there. Neither of them guessed who would win. Both worked from every way the rest of the game could have gone.

Counting the ways the game could end

The leader needs one more win. The other friend needs two. At most two more rounds would settle it. Write down every way those two rounds could go, where L is a round the leader wins and O a round the other friend wins:

• L then L: the leader wins.
• L then O: the leader wins (they won the first round).
• O then L: the leader wins.
• O then O: the other friend wins.

Four ways, equally likely if each round is a fair 50-50. The leader wins in three of them. So a fair split gives the leader three quarters, 75 points, and the other friend one quarter, 25 points.

Why count the second round even when the first one already decided the game? Because that makes all four ways equally likely, and only then can we simply count. That is the trick at the heart of probability.

Chance as a number

A probability is a number from 0 to 1.

• 0 means impossible: rolling a 7 on an ordinary die.
• 1 means certain: rolling a number from 1 to 6.
• Everything else sits between. A fair coin landing heads is 1/2, or 0.5, or 50%: three ways of writing the same number.

When every outcome is equally likely, the rule is simple:

probability = the number of ways the event can happen ÷ the number of ways anything can happen

The list of everything that can happen is called the sample space. For one die it is 1, 2, 3, 4, 5, 6. For two dice, it is every pair, from 1 and 1 to 6 and 6.

Two dice, thirty-six squares

Tap a total. The grid lights up every way two dice can make it. Then roll the dice many times and see whether the counts agree.

Why 7 is the most common total

Two dice have 6 × 6 = 36 equally likely pairs. Only one pair makes 2 (a 1 and a 1), so its chance is 1/36. Six pairs make 7: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. So its chance is 6/36, which is 1/6.

Is a 7 "due" after ten rolls without one? No: the dice do not remember. Is 7 more likely than 12 on every single roll? Yes: six ways against one. So what does probability tell you? Not what will happen next, but how often things happen when you try many times.

People often say a 50-50 chance means "it might or might not happen". But that is true of almost everything. A 50-50 chance is something much more exact: if you tried it many, many times, it would happen in about half of them.

Should schools teach probability earlier?

Probability is often taught late, after years of other mathematics. Should it come sooner?

A hard case. A student who leaves school at sixteen. Which would serve them better in life: more algebra, or more probability?

Explore in World 101

Chance is studied everywhere, from weather to medicine. Open Probability, Risk and Uncertainty in World 101 and follow one link that surprises you.