Experimental probability: relative frequency over many trials

0:00There are two honest ways to put a number on a chance, and they use completely different evidence. The first collects data. Run the thing many times, or find a record of many past runs, count how often the event happened, and divide by the number of attempts. That is experimental probability. The second never runs anything at all. It assumes every outcome is equally likely, counts the ones that qualify, and divides by the total.

0:34That is theoretical probability. One asks the world; the other reasons about the object. This video is about the first, and about the guarantee that makes it more than guesswork. Two words first, because the formula needs them. When you run an experiment, a result you could get is called an outcome. The collection of every outcome is the sample space, written inside curly brackets. A coin has a sample space of two: heads and tails.

1:09A die has six: one through six. Notice what a sample space is not. It is not a list of what happened - it is a list of what could happen, written down before you start. And nothing in it repeats. Each possibility appears exactly once, or the counting later would count it twice. Now the formula, which is almost too simple to state. The experimental probability of an event is the number of times it actually happened, divided by the number of trials you ran.

1:46Both of those numbers come straight off your record sheet. Nothing is assumed about the object. Not that it is fair, not that it is symmetric, nothing. That is the method's strength and its weakness in the same breath. Its strength, because it works on objects you cannot reason about at all. Its weakness, because the answer is only as good as the record, and a record is always finite.

2:14Here is a run. A die is rolled fifty times, and it lands on four exactly eight times. The experimental probability of a four, from that record, is eight over fifty. Which cancels to four twenty-fifths, or nought point one six, or sixteen per cent. Now set that beside what a fair die should give: one in six, about nought point one six seven. Eight out of fifty is a little under.

2:45The run has missed, by one part in a hundred and fifty. The usual next sentence is that experiments are unreliable. That sentence is wrong, and the next two minutes are why. Ask instead what fifty rolls could have given. Fifty rolls report a count of fours over fifty and nothing else, so the only answers available are nought over fifty, one over fifty, two over fifty, and so on up.

3:16A sixth of fifty is eight and a third, which is not a whole number. So no fifty-roll run lands on one sixth. Not an unlucky one - none of them. Search every count from nought to fifty and ask which comes closest, and the answer is eight. Nine misses by exactly twice as much. Nothing beats eight, and nothing even ties with it. That run was not unlucky; it was the best fifty rolls can do.

3:48A small point about names, because two of them mean the same thing. A statistician would call eight over fifty the relative frequency of a four: how often it happened, against how many chances it had. Experimental probability is the same quotient wearing a different label. These are not two quantities that happen to agree. They are one number, computed once, called two things depending on who is talking. If you have found one of them, you have found the other, and there is nothing further to do.

4:26That word could, from a moment ago, deserves its own scene, because it is the whole explanation. A run of any number of trials divides by that number, so the only values it can report are the multiples of one over it. Call that the grain of the instrument. Twelve rolls can only report twelfths. Roll a die twelve times, get a three on three of them, and your answer is a quarter - where one sixth is two twelfths, which twelve rolls can report exactly.

5:00So that run did not fall short of the truth. It overshot it, by a single occurrence. Twenty coin tosses, by contrast, can land on a half exactly, with ten heads, because twenty divides by two. More trials means a finer grain. A sixth of twelve is two, of sixty ten, of six hundred a hundred, of six thousand a thousand - every one a whole number, so every one of those runs can report a sixth exactly.

5:35And the grain goes from one twelfth to one six-thousandth: five hundred times finer. But a finer grain is not the guarantee. Here is the guarantee, counted rather than claimed. Toss a coin ten times, and missing a half by a tenth or more happens more often than it does not. At fifty tosses, about one time in five. At a hundred, about one in eighteen. At five hundred, better than one in a hundred thousand.

6:06At a thousand, better than one in a billion. Read that carefully, because it promises less than people hear. It does not say the experimental value will equal the theoretical one. Nothing obliges fifty rolls to come out right, and a million rolls do not oblige it either - a million is not a multiple of six, so a million rolls cannot report a sixth exactly. What many trials do is make a large miss wildly unlikely.

6:41Here is what they do not do. Watch one run of a hundred thousand tosses where heads stays a little ahead. At a hundred tosses, fifty-five heads: a proportion of nought point five five, and a lead of five. At a thousand, the lead is fifteen; at ten thousand, fifty. At a hundred thousand it is over a hundred and fifty - thirty times what it was - while the proportion has closed to within one part in six hundred of a half.

7:13Averaged over every possible run, the same story: quadruple the tosses and the expected gap doubles. The ratio settles down; the counts do not. So far the experiment has been a second opinion, checked against a theory we already had. Now take an object with no theory at all. Throw a paper cup in the air and it comes down one of three ways: standing on its bottom, upside down on its rim, or lying on its side.

7:48How likely is each? There is no way to find out by thinking. A die has six faces you can count and divide by; a cup has nothing of the kind, no symmetry, no equally likely anything. For a cup, trials are not a cheap substitute for the theory - they are the only access to the number there is. So throw it a hundred times and count. And notice something the record hands you free: whatever the three counts turn out to be, they add to a hundred, so the three probabilities add to one.

8:27One more distinction, and it is the one most often fumbled. Suppose fifty students are asked whether they like football, and fifteen say yes. Which of those numbers is the frequency, and which is the relative frequency? Fifteen is the frequency - a plain count of how many times something happened. The relative frequency is fifteen out of fifty, which is three tenths, or nought point three, or thirty per cent.

8:58The two are neither close nor interchangeable. A frequency can be any size at all; a relative frequency can never exceed one, because it is a part measured against its own whole. So what is an experimental probability worth? It is a measurement, and like every measurement it arrives with an error attached. That error has two separate sources. One is the grain: a run can only report multiples of one over its own length, and no amount of care gets round it.

9:35The other is the run itself: nothing forces any particular run to come out near the truth. More trials shrink both, and neither ever reaches nothing. And here is one thing that no number of trials ever buys. They say nothing whatever about the next one. Roll a four eight times running, and the ninth roll is still one in six. Count every four-roll sequence a die can produce: of the six that begin with three fours, exactly one ends in a fourth.

10:10The long run is what the number was ever about, and the next trial was never in it.