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15.4 Experimental and theoretical probabilities

15.4 Experimental and theoretical probabilities

One way to find probabilities is to use equally likely outcomes.
Another way is to collect data from a survey or an experiment. If you want to know the probability
that a seed will germinate, you cannot
Certain computer programs, called spreadsheets, have a
use equally likely outcomes. You need
random number generator that you can use to simulate
to plant seeds and count how many
throwing a dice.
germinate.
Here are the results of 20 throws of a dice simulated on a
computer.

5 2 5 1 2 6 2 5 3 1 3 3 5 1 1 1 1 3 3 3

Here are the results in a frequency table. Score 1 2 3 4 5 6

The relative frequencies (RF) give an
experimental probability for each number. Frequency 6 3 6 0 4 1

The theoretical probability for each number, Relative frequency 0.3 0.15 0.3 0 0.2 0.05
1
based on equally likely outcomes, is = 0.167.
6
The experimental and theoretical probabilities are not Relative frequency = frequency ÷ total throws
close. This is because 20 is a small number of throws.
What happens if you have more throws? Another set of throws will give different results.
This table shows the results after 20, 50, 100 and 200 throws.

20 throws 50 throws 100 throws 200 throws
Score
Frequency RF Frequency RF Frequency RF Frequency RF
1 6 0.3 11 0.22 19 0.19 36 0.18
2 3 0.15 11 0.22 19 0.19 36 0.18

3 6 0.3 8 0.16 19 0.19 35 0.175
4 0 0 5 0.1 15 0.15 31 0.155
5 4 0.2 7 0.14 13 0.13 29 0.145
6 1 0.05 8 0.16 15 0.15 33 0.165

Total 20 1 50 1 100 1 200 1
As the number of throws increases, the experimental probabilities get closer to the theoretical
probabilities. These results illustrate two important results.
1 If you repeat an experiment you get different results. Compare the first 10 throws with the second
10 throws to see this.
2 If you increase the number of times an experiment is repeated you generally get better estimates of
probabilities. 200 throws give more reliable results than 20 throws.
In realistic examples, such as the seed planting described above, we do not have theoretical probabilities
to compare with the experimental probabilities. That is why you need to do the experiment!
However, it remains true that different experiments will give different estimates, and you can have more
confidence in the estimates if the experiment is repeated more often.
Experimental probabilities are usually given as decimals or percentages.

154 15 Probability