7.2: Relative Frequency (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.
fr(E) = the number of times E occurs
Toss a coin 20 times. If heads comes up 13 times, then the frequency of the event that heads comes up is
fr(E) = 13.

The relative frequency or estimated probability of the event E is the fraction of times E occurs.

P(E) = \frac{fr(E)}{N}

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

P(E) = \frac{fr(E)}{N} = \frac{13}{20}

The number of times that the experiment is performed is called the sample size.
N = number of times the experiment is performed.
The experiment above was performed 20 times, so this is the sample size;
N = 20.

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
If 10 rolls of a die resulted in the outcomes 2, 1, 4, 4, 5, 6, 1, 2, 2, 1, then the associated relative frequency distribution is the one shown in the following table.

Outcome s	1	2	3	4	5	6
Frequency fr(s)	3	3	0	2	1	1
Relative Frequency P(s)	.3	.3	0	.2	.1	.1

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
(b) P(s_1) + P(s_2) + ... + P(s_n) = 1
(c) If E = \{\.e_1, .., e_r\.\}, then P(E) = P(e_1) + ... + P(e_r).
In words:
(a) The relative frequency of each outcome is a number between 0 and 1.
(b) The relative frequencies of all the outcomes add up to 1.
(c) The relative frequency of an event E is the sum of the relative frequencies of the individual outcomes in E.

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 and .

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Outcome	Dragon Quest	Star Pilot	Galactic Warrior 4	Detective III	Advanced Star Pilot
Relative Frequency

Q The relative frequency that the sum is 7 is P(E) =	.
	Answer should be accurate to 4 decimal places.