7.2: Estimated and Theoretical Probability
Part A: Estimated Probability
To start, here are some basic definitions.
Definition  Example 
The frequency of the event E is the number of times the event E occurs.

Toss a coin 20 times. If heads comes up 13 times, then the frequency of the event that heads comes up is

The relative frequency or estimated probability of the event E is the fraction of times E occurs.
Note: It follows that P(E) must be a number between 0 and 1 (inclusive). 
Referring to the situation above, the relative frequency of the event that heads comes up is

The number of times that the experiment is performed is called the sample size.

The experiment above was performed 20 times, so this is the sample size;

Note: If E happens to consist of a single outcome s, then we refer to P(E) as the relative frequency of the outcome s, and we write P(s).
Dice Simulation
To simulate the above experiment, press the "Throw Dice" button 30 times, or press the "Throw Dice 10 x" button three times. E is the event that the sum is 7, and its relative frequency will be calculated for you. You will see in the next tutorial that the modeled probability of E is 1/6 = .1666... The relative frequency should approach this number as the sample size gets large. You can now verify this experimentally. 
* Source: Technomic Inc./The New York Times, February 9, 1995, p. D4.
You can find more examples similar to those above in Section 7.2 of and .
Relative Frequency Distribution
The collection of the relative frequencies of all the outcomes is the relative frequency distribution or estimated probability distribution. Example

Following are some of the properties of relative frequency. Which one did you use in answering the last question?
Some Properties of Relative Frequency
Let S = \{\.s_1, s_2, ..., s_n\.\} be a sample space and let P(s_i) be the estimated probability of the event \{s_i\}. Then
(a) 0 &le P(s_i) ≤ 1
In words:

For more practice, try some of the questions in the chapter review exercises (Warning: it covers the whole of Chapter 7). Also try the exercises dealing with estimated probability in Section 7.2 of y .