Section 6.2: Introduction to Probability

Experimental 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 experimental probability of the event E is the fraction of times E occurs. P(E) = fraction of times E occurs = fr(E) N .	Referring to the situation above, the relative frequency 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 number of trials or the sample size. N = number of times the experiment is performed.	The experiment above was performed 20 times, so this is the sampe 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. The experimental probability of the outcome 7 is:

	30			4			2/15
	15/2

On-Line Simulation If your browser is Java-capable, press the Java button to bring up an applet that simulates this experiment and computes the experimental probability that the sum is 7.

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::

	0.4486			0.4809			0.5514
	0

* Source: Technomic Inc./The New York Times, February 9, 1995, p. D4.

You can find three more examples similar to those above on pages 405-407 in Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Notes

• The experimental probability of each outcome is always a number between 0 and 1.
• The experimental probabilities of all the outcomes add up to 1.
• We can obtain the experimental probability of an event E by adding up the experimental probabilities of the outcomes in E.

The theoretical probability, or simply probability, P(E), of an event E is a probability determined from the nature of the experiment rather than through actual experimentation.

Relationship to Experimental Probability: The experimental probability approaches the theoretical probability as the number of trials gets larger and larger.

Note From now on, when we refer to "probability," we will mean theoretical probability. When we want to refer to experimental probvability, we shall say so.

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.

Equally Likely Outcomes In an experiment in which all outcomes are equally likely, the 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 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 Now go back to the Java simulation of this experiment to see how closely the experimental probability approximates this number after a large number of trials.

Some advice from the Math Nerd...

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 1,287. The probability of being dealt a hand that does not consist entirely of red cards is approximately:

	0.0253			0.50			2,533,180
	0.9747

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:

	0.000495			0.02531			0.01957
	0.5

Probability Distribution A probability distribution is an assignment of a number P(si) to each outcome si in a sample space {s1, s2, . . . , sn}, so that 0 P(si) 1 and P(s1) + P(s2) + . . . + P(sn) = 1. In words: The probability of each outcome must be a number between 0 and 1, and the probabilities of all the outcomes must add up to 1. Given a probability distribution, we can obtain the probability of an event E by adding up the probabilities of the outcomes in E. If the probability of an event is zero, we call E an impossible event. The event Ø is always impossible, since something must happen.

Example (based on Example 7 on p. 414 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World ) A weighted dice has the probability distribution shown in the following table. (Notice that 6 is three times as likely to come up as any of the other numbers, and that the probabilities add up to 1). Outcome 1 2 3 4 5 6 Probability 0.125 0.125 0.125 0.125 0.125 0.375 The probability of an even number coming up is then P({2, 4, 6}) = 0.125 + 0.125 + 0.375 = 0.625. Therefore, there is a 62.5% chance that an even number will come up. Press the little icon for a Java simulation of this experiment.

The following is very similar to Exercise 43 on p. 419 in Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

The following table shows the week-end closing values of the Dow Jones Industrial Average (Dow) over the 10-week period beginning January 1, 1992:*

Week	1	2	3	4	5	6	7	8	9	10
Dow	3,210	3,200	3,270	3,220	3,220	3,250	3,280	3,270	3,220	3,240

* Source: The New York Times, July 18, 1993, Section 3, p.7. Averages are rounded to the nearest 10 points.

Low: The index is at or below 3,210
Middle: The index is above 3,210 but not above 3,250
High: The index is above 3,250

An appropriate experiment is:

	Note the week-end closing value of the Dow			Note whether the week-end closing value of the Dow is low, middle, or high
	Compute the probability that the week-end closing value of the Dow is low, middle, or high

Here is the table once again::

Week	1	2	3	4	5	6	7	8	9	10
Dow	3,210	3,200	3,270	3,220	3,220	3,250	3,280	3,270	3,220	3,240

Low: at or below 3,210
Middle: above 3,210 but not above 3,250
High: above 3,250

The experimental probability distribution is given by:

	High: 0.3; Middle: 0.5; Low: 0.2			High: 0.3265; Middle: 0.3235; Low: 0.3200
	High: 3; Middle: 5; Low: 2

For more practice, try some of the questions in the true/false quiz (warning: it covers the whole of chapter 6) by pressing the button on the sidebar. Then try the exercises on pp. 417-421 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.