7.2: Estimated and Theoretical Probability
Part A: Estimated Probability

Next tutorial: Part B: Theoretical 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).

The relative frequency that the sum is is P(E) = .
Answer should be accurate to 4 decimal places.
Now think about the set of outcomes when we throw a pair of distinguishable dice such as these.
.
Answer should be accurate to 4 decimal places.

You will see in the next tutorial that, when the outcomes are all equally likely (as they are here) then the fraction of outcomes that are in E is the modeled probability of E.
Dice Simulation To simulate the above experiment, press the various "Throw Dice" buttons until you have simulated at least 100 throws of the dice. Then compute the relative frequency of the event E that the sum is .
 
N (Sample size) Sum (Red + Green)
fr(E) (Frequency)       P(E) (relative frequency)
Answer should be accurate to 4 decimal places.

In 1993, there were approximately 10,000 fast food outlets in the US that specialized in Mexican food. Of these, the largest were Taco Bell with outlets, Taco John's with outlets and Del Taco with outlets.*

The relative frequency that a fast food outlet that specializes in Mexican food is is:

The relative frequency that a fast food outlet that specializes in Mexican food is is:

* Figures are slightly randomized for this tutorial but accurate to within 10 outlets. 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 s123456
Frequency fr(s) 3  3  0  2  1  1 
Relative Frequency P(s).3.3 0 .2.1.1

You run a commercial website that specializes in the sale of video games. The following statistics show the number of downloads of your five on-line video games last week.

GameDragon QuestStar PilotGalactic Warrior 4Detective IIIAdvanced Star Pilot
Downloads

 

Fill in the following relative frequency table and press "Check."

OutcomeDragon QuestStar PilotGalactic Warrior 4Detective IIIAdvanced Star Pilot
Relative Frequency

 

The estimated probability that a downloaded game is is .

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 relative frequency in Section 7.2 of y .

Last Updated: April, 2009
Copyright © 2009 Stefan Waner

Top of Page