Probability
Experimental probabilities
Drop a drawing pin and it lands point up or point down — but not equally often. When you cannot count equally likely outcomes, you run an experiment instead.
When counting will not work
Theoretical probability works when every outcome is equally likely — a fair dice, a fair spinner. A drawing pin is different. It can land point up or point down, but you have no reason to think those are equally likely. So you cannot calculate. You have to find out.
Relative frequency
Do the experiment many times. Each go is a TRIAL. relative frequency = frequency of the outcome ÷ total number of trials That relative frequency is your EXPERIMENTAL PROBABILITY — an estimate of the real one.
The drawing-pin experiment
124 drawing pins are dropped. 48 land point up and 76 land point down. P(point up) = 48 ÷ 124 = 0.39 to 2 d.p., so about 39%. P(point down) = 76 ÷ 124 = 0.61, so about 61%. Or just use 100% − 39% = 61%, since the two outcomes cover everything.
More trials, more reliable
Run the experiment again and you will not get exactly the same answer. That is normal. A LARGE number of trials gives a more reliable estimate than a small number. Ten flips of a fair coin could easily give 7 heads. A thousand flips will land much closer to half.
Worked examples
124 pins dropped, 48 point up
- Relative frequency = 48 ÷ 124.
- 48 ÷ 124 = 0.387…, which is 0.39 to 2 d.p.
Answer: About 39%
The same experiment, point down
- 76 ÷ 124 = 0.612…, which is 0.61 to 2 d.p.
- Or: 100% − 39% = 61%.
Answer: About 61%
400 cars, 140 over 60 km/h
- Relative frequency = 140 ÷ 400.
- 140 ÷ 400 = 0.35.
Answer: 0.35, or 35%
320 students, 16 travel by car
- 16 ÷ 320 = 0.05.
Answer: 0.05, or 5%
Stopped at the lights 32 days out of 50
- 32 ÷ 50 = 0.64.
- Not stopping: 1 − 0.64 = 0.36.
Answer: 0.64 to stop, 0.36 not to stop
Why do 1000 trials instead of 10?
- Every experiment varies a little.
- The more trials, the smaller that variation matters.
Answer: A large number of trials gives a more reliable estimate
Where you see this in real life
- Bus companies record how often each service runs late, then quote that as the chance of a delay.
- Drug trials give the medicine to thousands of people, because a handful of results would not be reliable.
Key words
- Experimental probability
- a probability estimated from the results of trials
- Relative frequency
- how often an outcome happened, out of all the trials
- Trial
- one go of an experiment
- Frequency
- the number of times an outcome happened
- Reliable
- likely to be close to the true value