7.2: Estimated and Theoretical Probability
Part A: Estimated Probability

Next tutorial: Part B: Theoretical Probability

(Based on Section 7.2 in Finite Mathematics and Finite Mathematics and Applied Calculus)

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)=fraction of times E occurs=
    fr(E)

    N
    		.

Note: It follows that P(E) must be a number between 0 and 1 (inclusive).

Referring to the situation above, the estimated probability of the event that heads comes up is

    P(E)=
    fr(E)

    N
    		=
    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.

 

A pair of dice (one red, one green) is cast 30 times, and on 4 of these occasions, the sum of the numbers facing up is 7.

Q The estimated probability of the outcome 7 is P(7) =

.
Answer should be accurate to 4 decimal places.

Dice Simulation
To simulate the above experiment, press the "Throw Dice" button 30 tims, or press the "Throw Dice 10 x" button three times.
 
   
N (Sample size) x (Red + Green)
fr(x=7) (Frequency)     P(x=7) (Estimated probability)

You will see later in the tutorial that the theoretical probability that x=7 is 1/6 = 0.1666... The estimated probability should approach this number as the sample size gets large. You can now verify this experimentally.

Q 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 4,809 outlets, Taco John's with 430 outlets and Del Taco with 275 outlets.* The experimental probability that a fast food outlet that specializes in Mexican food is none of the above is:

* 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 Finite Mathematics, or Finite Mathematics and Applied Calculus.

Probability Distribution

The collection of the estimated probabilities of all the outcomes is the estimated probability distribution or relative frequency 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 estimated probability distribution is the one shown in the following table.

Outcome123456
Probability0.30.3 0 0.20.10.1

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
Downloads12050155510

Q Fill in the following estimated probability table and press "Check."

OutcomeDragon QuestStar PilotGalactic Warrior 4Detective IIIAdvanced Star Pilot
Probability

Q The probability that a downloaded game is either Star Pilot or Advanced Star Pilot is .

   

Following are some of the properties of (estimated) probability. Which one did you use in answering the last question?

Some Properties of Estimated Probability

Let S = {s1, s2, ... , sn} be a sample space and let P(si) be the estimated probability of the event {si}. Then

(a) 0 P(si) 1
(b) P(s1) + P(s2) + ... + P(sn) = 1
(c) If E = {e1, e2, ..., er}, then P(E) = P(e1) + P(e2) + ... + P(er).

In words:
(a) The estimated probability of each outcome is a number between 0 and 1.
(b) The estimated probabilities of all the outcomes add up to 1.
(c) The estimated probability of an event E is the sum of the estimated probabilities of the individual outcomes in E.

For more practice, try some of the questions in the chapter quiz (Warning: it covers the whole of Chapter 7) by pressing the button on the sidebar. Then try the exercises dealing with estimated probability in Section 7.2 of Finite Mathematics and Finite Mathematics and Applied Calculus

Last Updated: June, 2006
Copyright © 1999, 2003, 2006 Stefan Waner

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