7.2: Estimated and Theoretical Probability
Part B: Theoretical Probability

Previous tutorial: Part A: Estimated Probability

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

Theoretical Probability

The theoretical probability, or probability, P(E), of an event E is the fraction of times we expect E to occur if we repeat the same experiment over and over.

Relationship with Estimated Probability
The estimated probability approaches the theoretical probability as the number of trials gets larger and larger. Thus,

Estimated probability is an approximation, or estimate, of theoretical probability. The larger the number of trials, the more accurate we expect this approximation to be.

If E consists of a single outcome s, we refer to P(E) as the probability of the outcome s, and write P(s) for P(E). The collection of the probabilities of all the outcomes is the probability distribution.

Determining Theoretical Probability
Theoretical probability is determined analytically, that is, by using our knowledge about the nature of the experiment rather than through actual experimentation. The best we can obtain through actual experimentation is an estimate of the theoretical probability (hence the term "estimated probability").

Examples
1. Toss a fair coin and observe the uppermost side. Since we expect that heads is as likely to come up as tails, we conclude that the theoretical probability distribution is

P(H) = 1/2, P(T) = 1/2.

2. Roll a fair die. Since we expect to roll a "1" one sixth of the time,

P(1) = 1/6.
Similarly, P(2) = 1/6, P(3) = 1/6, . . . , P(6) = 1/6.

3. Roll a pair of fair dice (recall that there are a total of 36 outcomes if the dice are distinguishable). If E is the event that the sum of the uppermost numbers is 5, then E = {(1, 4), (2, 3), (3, 2), (4, 1)}. Since all 36 outcomes are equally likely,

P(E) =

 
Note From now on, "probability," we will refer to either estimated or theoretical probability, depending on the context.

Calcuating theoretical proability requires a detailed knowledge of the experiment you are considering. A simple kind of experiment is one where the outcomes are equally likely. That is, where each outcome is as likely to occur as any other.

Calculating Theoretical Probability When Outcomes Are Equally Likely

In an experiment in which all outcomes are equally likely, the theoretical probability of an event E is

    P(E) =
    Number of favorable outcomes

    Total number of outcomes
    		=
    n(E)

    n(S)
    		.

(The "favorable outcomes" are the outcomes in E.)

Example

A pair of dice (one red, one green) is cast. We find the theoretical probability that the sum of the numbers facing up is 7.

    S = sample space = set of all pairs of numbers 1 through 6
    n(S) = 36
(Go back to the beginning of the previous tutorial if you need a refresher on this...)

    E = set of favorable outcomes = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
    n(E) = 6

    P(E) =
    n(E)

    n(S)
    		 =
    6

    36
    		 =
    1

    6

Invoke the on-line simulation of this experiment to see how closely the estimated probability approximates this number after a large number of trials.

You are dealt a hand of five cards from a standard deck of playing cards. The number of possible hands is 2,598,960. The number of possible hands consisting entirely of red cards (diamonds and hearts) is 65,780, and the number of possible hands consisting entirely of diamonds is 1287.

Q The probability of being dealt a hand that does not consist entirely of red cards is approximately:

Q If you are dealt a hand of five red cards from a standard deck of playing cards, the probability of having a hand consisting entirely of diamonds is approximately:

Just as in the case of estimated probability, we have the following for theoretical probability.
Probability Distribution
The collection of the theoretical probabilities of all the outcomes is the theoretical probability distribution..

Example
A certain weighted die has the following theoretical probability distribution.

Outcome  1    2    3    4    5    6  
Probability.2.2 .2  .2.1.1

Notice that this die is twice as likely to show 5 or to show 6 than any of 1, 2, 3, or 4.

You toss a fair coin 3 times, and note the number of heads that comes up.

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

Outcome (Number of Heads)0123
Probability

Q The probability that either one or two heads will come up is.

   

Just as with estimated probability, the following rules hold for theoretical probability Which one did you use in answering the last question?

Some Properties of Theoretical Probability

Let S = {s1, s2, ... , sn} be a sample space and let P(si) be the theoretical 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 theoretical probability of each outcome is a number between 0 and 1.
b. The theoretical probabilities of all the outcomes add up to 1.
c. The theoretical probability of an event E is the sum of the theoretical probabilities of the individual outcomes in E.

Thus, estimated and theoretical probability satisfy the same general properties, so they are both known as "probability distributions." Properties of probability distributions are discussed in the next tutorial.

A Note on Populations and Samples

Q Last year, 450 of my 500 customers purchased one or more of my on-line video clips. Thus, the probability that a randomly selected customer purchased an on-line video clip is 450/500 = 0.9. Is this estimated probability or theoretical probability?
A We are given all the information we need to calculate the "actual" or theoretical probability, since we know the actual proportion of customers who purchased the video clips. On the other hand, if the scenario was

"Last year, 450 of 500 surveyed customers purchased one or more of my on-line video clips,"

then we would not know thte actual proportion, since there may be more customers than were surveyed. Thus, the probability would be estimated (being based on a sample, rather than the entire population).,

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 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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