Experimental Probability

Doing the experiment

A fair coin lands heads with probability $\frac{1}{2}$1/2, so in 40 throws you might expect 20 heads. Throw a coin 40 times and count, though, and the result is usually a little different. In one run of 40 throws, the coin came up heads 23 times and tails 17 times.

23 heads and 17 tails from 40 throws. The two bars are close, but they are not level.

A chance you measure

Each throw is called a trial. Heads came up in 23 of the 40 trials, so the fraction of the throws that were heads is $\frac{23}{40}$23/40. A probability found this way, by doing an experiment and counting, is called the experimental probability: the number of times the event happened ÷ the number of trials.

The probability worked out by counting equally likely outcomes, $\frac{1}{2}$1/2 for heads on a coin, is called the theoretical probability. $\frac{23}{40}$23/40 is a little more than $\frac{20}{40}$20/40, which is $\frac{1}{2}$1/2. The experimental probability is close to the theoretical probability, but it is not equal to it.

One part is one throw, and the 23 heads are colored. The dashed line marks half of the throws, 20, and the heads run 3 past it.

Another run, another answer

Throw the coin 40 more times and the count changes: perhaps 19 heads this time, and 22 the time after. Each run gives its own experimental probability, $\frac{19}{40}$19/40 or $\frac{22}{40}$22/40, and none of them has to be exactly $\frac{1}{2}$1/2. Chance makes every run a little different.

Three runs of 40 throws gave 23, 19 and 22 heads. Each one lands near the half line, and each one misses it by a different amount.

When there is nothing equal to count

For a coin or a dice, the theoretical probability comes from counting equally likely outcomes. Some things have no equally likely outcomes to count. A thumbtack can land point up or point down, but the two are not equally likely, and nothing about its shape says by how much. The way to find the chance is to do the experiment.

Toss a thumbtack 50 times. If it lands point up 32 times, the experimental probability of point up is $\frac{32}{50}$32/50, which is $\frac{64}{100}=0.64$64/100 = 0.64.

One part is one toss of the thumbtack. The 32 tosses that landed point up are colored: $\frac{32}{50}=0.64$32/50 = 0.64.

All the trials go on the bottom

The bottom number is every trial, never only the misses. 23 heads against 17 tails gives $\frac{23}{17}$23/17, which is more than 1, and no probability is more than 1. Heads out of all the throws is $\frac{23}{40}$23/40.

The trials do not go on top either. $\frac{40}{23}$40/23 is also more than 1. The number of times the event happened goes on top.