The theoretical probability of an event is the ratio of the number of favorable outcomes to the number of all possible outcomes. Keep in mind that when calculating the theoretical probability, all outcomes in a sample space must be equally likely.
P(Event)=Number of Favorable Outcomes/Number of Possible Outcomes
For example, consider rolling a standard six-sided die. There are 6 equally likely outcomes in the sample space.
For the event of rolling an odd number, there are 3 favorable outcomes. Therefore, the probability of the event is calculated as follows.
P(Odd Number) = 3/6 = 50 %
Note that we used a probability model with equally likely outcomes in the sample space to find the probability.
Experimental Probability
The experimental probability of an event is the ratio of the number of times the event occurs to the number of times an experiment is done. It is based on the data collected from repeated trials of the experiment.
P(Event) = Number of Times Event Occurs/Number of Trials
Let's again consider rolling a standard six-sided die, but this time we will collect the data by repeating trials to find the probability. To do so, we will roll a die 30 times.
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2 & 1 & 2 & 4 & 1 & 5
1 & 1 & 4 & 2 & 1 & 4
1 & 5 & 1 & 2 & 6 & 3
6 & 5 & 6 & 4 & 1 & 2
4 & 5 & 1 & 3 & 5 & 5
Keep in mind that this is only one example of data collected by rolling a die. There are 17 outcomes that are odd numbers. We can now calculate the experimental probability of rolling an odd number.
P(Odd Number) = 17/30 ≈ 57 %
When calculating an experimental probability, even outcomes that have equally likely theoretical probabilities can end up being not equally likely. For example, someone could replace the die with a fake die consisting of a second 1 instead of a 2.
Differences
Finally, we can conclude the differences between theoretical probability and experimental probability.
Differences
Theoretical Probability
Experimental Probability
Based on a probability model
Based on an experiment
Requires equally likely outcomes in a sample space
