Copyright © 2016, 2013, and 2010, Pearson Education, Inc. 9 Chapter Probability Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

9-1 Determining Probabilities Students will be able to understand and explain: How probabilities are determined; Experimental or empirical probabilities versus theoretical probabilities; Properties of probabilities; Mutually exclusive and non-mutually exclusive events; and Geometric probabilities.

Definitions Experiment: an activity whose results can be observed and recorded. Outcome: each of the possible results of an experiment. Sample space: a set of all possible outcomes for an experiment. Event: any subset of a sample space.

Example Suppose an experiment consists of drawing 1 slip of paper from a jar containing 12 slips of paper, each with a different month of the year written on it. Find each of the following: a. the sample space S for the experiment S = {January, February, March, April, May, June, July, August, September, October, November, December}

Example (continued) b. the event A consisting of outcomes having a month beginning with J A = {January, June, July} c. the event B consisting of outcomes having the name of a month that has exactly four letters B = {June, July}

Example (continued) d. the event C consisting of outcomes having a month that begins with M or N C = {March, May, November}

Determining Probabilities Experimental (empirical) probability: determined by observing outcomes of experiments. Theoretical probability: the outcome under ideal conditions. Equally likely: when one outcome is as likely as another Uniform sample space: each possible outcome of the sample space is equally likely.

Law of Large Numbers (Bernoulli’s Theorem) As the number of trials of an experiment increases, the experimental or empirical probability of a particular event approaches a fixed number, the theoretical probability of that event.

Probability of an Event with Equally Likely Outcomes For an experiment with non-empty finite sample space S with equally likely outcomes, the probability of an event A is

Example Let S = {1, 2, 3, 4, 5, …, 25}. If a number is chosen at random, that is, with the same chance of being drawn as all other numbers in the set, calculate each of the following probabilities: a. the event A that an even number is drawn A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}, so n(A) = 12.

Example (continued) b. the event B that a number less than 10 and greater than 20 is drawn c. the event C that a number less than 26 is drawn C = S, so n(C) = 25.

Example (continued) d. the event D that a prime number is drawn D = {2, 3, 5, 7, 11, 13, 17, 19, 23}, so n(D) = 9. e. the event E that a number both even and prime is drawn E = {2}, so n(E) = 1.

Definitions Impossible event: an event with no outcomes; has probability 0. Certain event: an event with probability 1.

Probability Theorems If A is any event and S is the sample space, then The probability of an event is equal to the sum of the probabilities of the disjoint outcomes making up the event.

Example If we draw a card at random from an ordinary deck of playing cards, what is the probability that a. the card is an ace? There are 52 cards in a deck, of which 4 are aces.

Example (continued) If we draw a card at random from an ordinary deck of playing cards, what is the probability that b. the card is an ace or a queen? There are 52 cards in a deck, of which 4 are aces and 4 are queens.

Mutually Exclusive Events Events A and B are mutually exclusive if they have no elements in common; that is, For example, consider one spin of the wheel. S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0, 1, 2, 3, 4}, and B = {5, 7}. If event A occurs, then event B cannot occur.

Mutually Exclusive Events If events A and B are mutually exclusive, then The probability of the union of events such that any two are mutually exclusive is the sum of the probabilities of those events.

Complementary Events Two mutually exclusive events whose union is the sample space are complementary events. For example, consider the event A = {2, 4} of tossing a 2 or a 4 using a standard die. The complement of A is the set A = {1, 3, 5, 6}. Because the sample space is S = {1, 2, 3, 4, 5, 6},

Complementary Events If A is an event and A is its complement, then

Non-Mutually Exclusive Events Let E be the event of spinning an even number. E = {2, 14, 18} Let T be the event of spinning a multiple of 7. T = {7, 14, 21}

Summary of Probability Properties 1. P(Ø) = 0 (impossible event) 2. P(S) = 1, where S is the sample space (certain event). 3. For any event A, 0 ≤ P(A) ≤ 1.

Summary of Probability Properties 4. If A and B are events and A ∩ B = Ø, then P(A U B) = P(A) + P(B). 5. If A and B are any events, then P(A U B) = P(A) + P(B) − P(A ∩ B). 6. If A is an event, then P(A) = 1 − P(A).

Example A golf bag contains 2 red tees, 4 blue tees, and 5 white tees. a. What is the probability of the event R that a tee drawn at random is red? Because the bag contains a total of 2 + 4 + 5 = 11 tees, and 2 tees are red,

Example (continued) b. What is the probability of the event “not R”; that is a tee drawn at random is not red? c. What is the probability of the event that a tee drawn at random is either red (R) or blue (B); that is, P(R U B)?

Other Views of Probability Alternative Definition of Probability of an Event from Sample Space with Equally Likely Outcomes

Geometric Probability (Area Models) A probability model that uses geometric shapes is an area model. When area models are used to determine probabilities geometrically, outcomes are associated with points chosen at random in a geometric region that represents a sample space. This process is referred to as finding geometric probabilities.

Example Design a geometric model for the following experiments. a. Tossing a fair coin. b. Rolling a fair die.