Experimental Probability and Relative Frequency

There are two kinds of probability. Theoretical probability is worked out by counting — a coin has one head out of two sides, so the probability is one half, and you never have to toss anything. Experimental probability is measured by actually doing it: toss a coin 60 times, get 33 heads, and the experimental probability is 33 over 60, which is 0.55. That is also called the relative frequency, and the formula is frequency over total trials. The two answers differ, 0.5 against 0.55, and neither is wrong — they measure different things. Theoretical says what should happen; experimental says what did happen. Read the question to see which one it wants.

The more trials you run, the closer the relative frequency gets to the theoretical probability. Ten tosses might give 7 heads, a relative frequency of 0.7 — a long way from 0.5. Sixty tosses gave 0.55. Five hundred might give 0.522, and five thousand around 0.503. It is not that the coin corrects itself; a coin has no memory, and later tosses do not make up for earlier ones. It is simply that a few unusual results matter less and less as the total grows. The practical consequence is that a small experiment proves nothing about whether a coin or die is fair.

A probability lets you predict how many times something will happen. The expected number is the probability multiplied by the number of trials. Rolling a die 300 times, you expect one sixth of 300, which is 50 sixes. Sixty coin tosses give an expected 30 heads. Forty card draws with replacement give an expected 10 hearts. One important caution about the word expected: it means the long-run average, not a guarantee. You will rarely get exactly 50 sixes in 300 rolls, and 44 or 57 would be perfectly normal. It is the best single prediction you can make, not a promise about what will occur.

A common exam question asks whether a die or spinner is fair. The method is to work out what you would expect, then compare. Six hundred rolls of a fair die should give one sixth of 600, so about 100 of each face. Results of 98, 103, 96 and 101 are all close to 100 — that is normal variation and no cause for suspicion. But 55 and 147 are a long way from 100 over that many rolls, so the die appears biased towards 6 and against 5. Note the wording: say 'appears' or 'the results suggest', because an experiment gives evidence, not proof. And the sample must be large — four faces out of a dozen rolls tells you nothing.

Sometimes theory simply cannot help. Use theory when outcomes are equally likely and countable — a fair die, a fair coin, a shuffled deck, a spinner with equal sectors. Use an experiment when they are not. A drawing pin has two outcomes, point-up or point-down, but they are clearly not equally likely, so you cannot claim one half just because there are two possibilities. The only way to find out is to drop a lot of pins and count. The same applies to a bus arriving late, a player scoring a free throw, or a machine producing a faulty part. That is why insurers, sports analysts and factories all rely on relative frequency — there is no theory to count, only data to collect.