Theoretical and Experimental Probability

1 · Theoretical probability: what should happen

Some probabilities you can work out just by thinking. A normal dice has six faces, 1 to 6. If it's fair (not damaged or weighted), each face is just as likely to land on top as any other. We say the six results are equally likely.

A few words you'll need all the way through this page:

When all the outcomes are equally likely, count them:

P(event) = number of favourable outcomes ÷ total number of outcomes

("Favourable" just means "the ones that count for your event".) You met this idea on the probability scale page. Tap the dice faces below to build an event, or pick one from the buttons.

Notice that theoretical probability tells you what is likely, not what will happen. P(six) = 1/6 does not mean "every sixth roll is a six". To see what really happens, we need to do an experiment.

2 · Experiments, trials and relative frequency

An experiment is something you do to collect results, like flipping a coin lots of times. Each single flip is called a trial. As you go, you count how many times each outcome happens. That count is the frequency (you might record it with a tally).

A frequency on its own doesn't tell you much. "I got 7 heads" sounds very different if you flipped 10 times or 1000 times. So we divide by the number of trials. This gives the relative frequency: the fraction of the trials on which the event happened.

relative frequency = frequency ÷ number of trials

Relative frequency is also called experimental probability, because it's a probability you find from an experiment. Like any probability it is between 0 and 1. Have a go: flip the coin and watch the table fill up.

The relative frequencies of heads and tails always add up to 1, because every flip was one or the other. (That's the same rule as P(not A) = 1 − P(A); see mutually exclusive outcomes.)

3 · More trials, better estimates

Here is the big idea of this whole topic. Roll a fair dice and keep track of the relative frequency of sixes after every single roll. Theory says P(six) = 1/6 ≈ 0.167. Watch the orange line (your experiment) and the dashed teal line (theory).

Try several runs (press New run, and the old one turns grey). At the start, every run zig-zags wildly: one lucky six out of 3 rolls gives a relative frequency of 0.333, twice the theory. But as the number of trials grows, each run settles down close to the theoretical value. It never has to land exactly on it, but it gets closer and closer. Mathematicians call this the law of large numbers.

Same experiment, different groups

Imagine 30 groups in a school each flipping a coin. Does every group get the same relative frequency of heads? Choose how many flips each group does, then compare. Each dot is one group's result.

With 10 flips, results are spread right across the scale: some groups get 0.2 or 0.8 just by chance. With 1000 flips, every group lands very close to 0.5. Small experiments vary a lot; big experiments agree. That's why a scientist would never trust an estimate from only a handful of trials.

4 · When outcomes aren't equally likely

Drop a drawing pin (a thumb tack) on a table. It lands either point up (resting on its flat head) or point down (on its side, with the point touching the table). Are these equally likely? No! The shape of the pin makes one more likely, and there's no way to count "equally likely outcomes" to work it out. Theoretical probability can't help.

The only way is an experiment: drop it lots of times and use the relative frequency as an estimate of the probability. This pin has a secret probability chosen by the page. Drop it, then estimate P(point up).

The same idea works for real-world data. Insurance companies, weather forecasters and doctors all estimate probabilities from what happened before: relative frequency = number of times it happened ÷ number of times we looked. Practise with real-looking data:

5 · Is it fair? Spotting a biased dice

A dice (or coin, or spinner) is biased if some outcomes are more likely than others, for example because it's been weighted on one side. You can't tell by looking. But you can test it: roll it many times and compare each relative frequency with the 1/6 ≈ 0.167 that a fair dice should give.

First, make your own dice: drag the plum bars to make some faces more likely, then roll and see how the bias shows up. Then switch to mystery dice: the page secretly picks a fair or biased dice, and you decide from the evidence.

Three things to remember when you judge fairness:

• Expect some wobble. Even a perfectly fair dice won't give exactly 1/6 for every face. Small differences are normal.
• Use lots of trials. With 30 rolls, a fair dice can look biased and a biased dice can look fair. With 1000 rolls, a real bias stands out clearly.
• You can never be 100% sure. An experiment gives evidence, not proof. More trials make you more confident.

6 · Expected frequency

Now turn the question round. If you know (or have estimated) a probability, how many times would you expect the event to happen in a certain number of trials? If P(six) = 1/6 and you roll 60 times, about one roll in every six should be a six, so you'd expect about 60 ÷ 6 = 10 sixes. That's the expected frequency:

expected frequency = probability × number of trials

Expected frequency is a prediction of what happens on average, not a promise. Rolling 60 times you might get 8 sixes, or 13. And it doesn't have to be a whole number: in 50 rolls you'd expect 50 × 1/6 = 8.33… sixes. Use the calculator, then press Do the trials to compare with a real (simulated) experiment.

Practice: expected frequency

Each question is new. If you go wrong, the feedback tries to spot which mistake you made.

7 · Common mistakes: streaks, "exactly half" and too few trials

Mistake 1: "It's due a six"

You've rolled a dice ten times and not seen a six. Surely a six is due now? No. A dice has no memory. Every roll is a fresh start, and P(six) is still 1/6. In the same way, after five heads in a row a fair coin is no more likely to land tails. Believing otherwise is called the gambler's fallacy (a fallacy is a mistaken belief), and it has cost gamblers a lot of money.

Test it yourself. The page flips thousands of coins, finds every time there was a streak of heads, and looks at what came next.

Mistake 2: "100 flips means exactly 50 heads"

Expected frequency of heads in 100 flips is 1/2 × 100 = 50. But how often do you get exactly 50? Make a prediction, then flip 100 coins many times and see how the totals spread out.

Exactly 50 heads happens only about 8% of the time (roughly 1 time in 12). But a result between 40 and 60 happens about 96% of the time. So 50 is the most likely single total, and results cluster round it, but "exactly 50" is unlikely. Getting 47 or 54 heads is no evidence at all that a coin is biased.

Worked examples

Quick check

What you've learned

• Theoretical probability is worked out without an experiment. When outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. For a fair dice, P(six) = 1/6.
• An experiment is made of trials. The frequency is how many times something happened.
• Relative frequency (experimental probability) = frequency ÷ number of trials. It is always between 0 and 1, and the relative frequencies of all the outcomes add up to 1.
• Small experiments vary a lot. As the number of trials increases, the relative frequency settles down towards the theoretical probability, so more trials give a more reliable estimate.
• When outcomes are not equally likely (drawing pins, biased dice, real-world events), use relative frequency from lots of trials or past data to estimate the probability.
• To decide whether a dice or coin is fair, compare relative frequencies with the theoretical values over many trials. Small differences are normal; large, consistent ones suggest bias.
• Expected frequency = probability × number of trials. It's an average prediction, not a guarantee, and it needn't be a whole number.
• Dice and coins have no memory: nothing is ever "due" (the gambler's fallacy). And 100 flips rarely give exactly 50 heads.