Illinois State University Mathematics Department


MAT 312: Probability and Statistics for Middle School Teachers

Spring 1999
9:35 - 10:50 am TR STV 350A
Dr. Roger Day (day@math.ilstu.edu)



Discussion Notes: Session #23
Tuesday 6 April 1999

Assignment Due Today
Useful Terms, Symbols, and Properties
(continued from
Session #21)

Probability Problems

Assignment for Next Time

Assignment Due Today
  • Review the counting techniques we've learned during previous sessions.
  • Read the Content Background of Chapter 2 in the Teacher's Guide textbook.
  • Study the differences between experimental and theoretical probabilities.
  • Complete the tasks associated with four experiments.
  • Become familiar with probability terms, symbols, and properties. For as many as possible and appropriate, you should have at least one example to help explain and clarify the term, symbol, or property.
  • Practice on the Quiz #4 Sample in preparation for the real Quiz #4 next week.

Terms, Symbols, and Properties

So that we can be efficient and clear in our discussions and calculations associated with probability, we will identify some terminology and symbolism to help us. We also point out some fundamental properties of probability.

outcomes: the possible results of an experiment

equally likely outcomes: a set of outcomes that each have the same likelihood of occurring.

sample space: the set of all possible outcomes to an experiment.

uniform sample space: a sample space filled with equally likely outcomes.

nonuniform sample space: a sample space that contains two or more outcomes that are not equally likely.

event: a collection of one or more elements from a sample space.

expected value: the long-run average value of the outcome of a probabilitic situation; if an experiment has n outcomes with values a(1), a(2), . . . , a(n), with associated probabilities p(1), p(2), . . . , p(n), then the expected value of the experiment is

a(1)*p(1)+ a(2)*p(2) + . . . + a(n)*p(n).

There is further discussion of the use and computation of expected value.

random event: an experimental event that has no outside factors or conditions imposed upon it.

P(A): represents the probability P for some event A.

probability limits: For any event A, it must be that P(A) is between 0 and 1 inclusive.

probabilities of certain or impossible events: An event B certain to occur has P(B) = 1, and an event C that is impossible has P(C) = 0.

complementary events: two events whose probabilities sum to 1 and that share no common outcomes. If X and Y are complementary events, that P(A) + P(B) = 1.

mutually exclusive events: two events that share no outsomes. If events C and D are mutually exclusive, then P(C or D) = P(C) + P(D); if two events are not mutually exclusive, then P(C or D) = P(C) + P(D) - P(C and D).

independent events: two events whose outsomes have no influence on each other. If E and F are independent events, than P(E and F) = P(E) * P(F).

conditional probability: the determination of the probability of an event taking into account that some condition may affect the outcomes to be considered. The symbol P(A|B) represents the conditional probability of event A given that event B has occurred. Conditional probability is calculated as P(A|B) = P(A and B)/P(B).

geometrical probability: the determination of probability based on the use of a 1-, 2-, or 3-dimensional geometric model.

Expected Value

Expected value is the long-run average value of the outcome of a probabilitic situation. If an experiment has n outcomes with values a(1), a(2), . . . , a(n), with associated probabilities p(1), p(2), . . . , p(n), then the expected value of the experiment is a(1)*p(1)+ a(2)*p(2) + . . . + a(n)*p(n).

For example, if the experiment is to roll a fair die and record the output showing on the top face of the die, the outcomes are {1,2,3,4,5,6}. Because each outcome is equally likely, each outcome has probability of 1/6. The expected value for this experiment is the weighted average of the outcomes and the probabilities, calculated using the summation shown above:

(1)*(1/6) + (2)*(1/6) + (3)*(1/6) + (4)*(1/6) + (5)*(1/6) + (6)*(1/6) = (1+2+3+4+5+6)*(1/6) = 21/6 = 3.5

The expected value provides us information about what to expect is the experiment is carried out many many times. Here, the average output we will get is 3.5. Note that 3.5 is not one of the possible outcomes of the experiment, but rather a weighted average, representing what a "typical" outcome will be over the long-term running of the experiment.

Expected value is a useful concept to turn to when evaluating whether a game of chance is fair. Suppose that Allen and Zenda play a game with a die, each rolling the die on alternate turns. When someone rolls a 5 or 6, Allen wins as many points as are shown on the die. When someone rolls a 1,2,3, or 4, Zenda wins as many points as are showing on the die. The first player to accumulate 25 points or more is the winner. What is the expected value for each player?

For Allen, the expected value is (5)*(1/6) + (6)*(1/6) = 11/6. For Zenda, it is (1)*(1/6) + (2)*(1/6) + (3)*(1/6) + (4)*(1/6) = 10/6. This is not a fair game, for in the long run, Allen will win more points per play than Zenda.

 

Here's another example.

There are four possible outcomes when a gambler plays the spinner game shown to the right. The green area, $50, represents 1/2 the circle, the violet area, $20, represents 1/3 of the circle, and each of the other areas ($10 and $0) is 1/12 of the circle.

A player spins the spinner and wins the amount upon which the spinner lands.

What should the player pay so that this is a fair game?

The expected winnings are

(50)(1/2) + (20)(1/3) + (10)(1/12) + (0)(1/12), which equals $32.50.

If that is what a player can expect to win per play, over the long haul, then to make this a fair game the player should pay $32.50 per spin. Any less than that and the player has an advantage. Any more than that and the game owner will earn a profit, again, over many many plays of the game.

Before you go to Las Vegas to win your fortune, be sure to calculate the expected value of the games you'll be playing!

Assignment for Next Time