Probability
Probability from experiments
Work out the empirical probability of an event from the results a trial actually gave.
There are two ways to answer "how likely is it?". One is to reason it out: a fair coin has two sides, so a head has probability one half. The other is to go and try it — toss the coin two hundred times and count. The second kind is called empirical probability, and it is the only kind available for questions nobody can reason about, like whether a drawing pin lands point up, or whether a particular bus is late.
The definition
Empirical probability = the number of times the event happened, divided by the number of times the trial was run. Nothing about fairness or symmetry comes into it. It is a record of what did happen, not a prediction of what should.
Worked example
In 40 tosses a drawing pin landed point up 26 times. What is the empirical probability of point up?
The event happened 26 times, out of 40 trials. So the probability is 26/40.
Trials on the bottom, successes on the top — the same way round as any fraction of a whole.
Have a play
Every probability there has ever been lives somewhere on this line. Move along it and read off the values.
Tap anywhere on the line.
Try it together
A drawing pin was tossed 250 times and landed point up 150 times. Find the empirical probability of point up.
Two numbers are needed: how often it happened, and how often it was tried.
1.How many times did the pin land point up?
More trials, steadier answer
Toss a fair coin ten times and seven heads would not be strange at all. Toss it a thousand times and the fraction of heads settles close to one half and stays there. Empirical probability is only worth trusting when the trial has been run enough times, which is why experiments in this topic run into hundreds.
Have a go
Have a go on your own: in 80 matches a team won 20. What is the empirical probability that it wins? Give your answer as a decimal.
Hint: Divide the wins by the matches.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.