## THE RELATIVE FREQUENCY INTERPRETATION OF PROBABILITY

We are interested in learning about the probability of some event in some process. For example, our process could be rolling two dice, and we are interested in the probability in the event that the sum of the numbers on the dice is equal to 6.

Suppose that we can perform this process repeatedly under similar conditions. In our example, suppose that we can roll the two dice many times, where we are careful to roll the dice in the same manner each time.

I did this dice experiment 50 times. Each time I recorded the sum of the two dice and got the following outcomes:

```    4   10    6    7    5   10    4    6    5    6   11   11    3    3    6
7   10   10    4    4    7    8    8    7    7    4   10   11    3    8
6   10    9    4    8    4    3    8    7    3    7    5    4   11    9
5    5    5    8    5
```

To approximate the probability that the sum is equal to 6, I count the number of 6's in my experiments (5) and divide by the total number of experiments (50). That is, the probability of observing a 6 is roughly the relative frequency of 6's.

```
# of 6's
PROBABILITY (SUM IS 6) is approximately -----------
# of tosses

5
=     ----  =  .1
50

```

In general, the probability of an event can be approximated by the relative frequency , or proportion of times that the event occurs.

```                                             # of times event occurs
PROBABILITY (EVENT) is approximately     -----------------------
# of experiments
```

Comments about this definition of probability:

1. The observed relative frequency is just an approximation to the true probability of an event. However, if we were able to perform our process more and more times, the relative frequency will eventually approach the actual probability. We could demonstrate this for the dice example. If we tossed the two dice 100 times, 200 times, 300 times, and so on, we would observe that the proportion of 6's would eventually settle down to the true probability of .139.

Click here for a demonstration of this idea using computer dice rolling.

2. This interpretation of probability rests on the important assumption that our process or experiment can be repeated many times under similar circumstances. In the case where this assumption is inappropriate, the subjective interpretation of probability is useful.

Page Author: Jim Albert (© 1996)
albert@bayes.bgsu.edu
Document: http://www-math.bgsu.edu/~albert/m115/probability/relfreq.html
Last Modified: November 24, 1996