1. Probability 9.8

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Definition: Experiment Any activity with an unpredictable results is called an experiment. The results of an experiment are called outcomes and the set of all possible outcomes is the sample space. Examples: Identify the sample space. Flip a coin. Toss a die. S = {H, T} S = {1, 2, 3, 4, 5, 6} The number of outcomes in the sample space S is n(S). 2 6 ExperimentSample Spacen(S)n(S)

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Any subset of the sample space is called an event. Examples: List the outcomes in each event. Flip a coin Toss a die Draw a card Flip two coins Examples: List Outcomes The number of outcomes in an event E is n(E). 3 4 4 3 Get heads{H} Get an even number{2, 4, 6} Get a 3 or higher{3, 4, 5, 6} Get an 8 {  8, 8,  8,  8} Get at least one head{HH, HT, TH} ExperimentEventn(E)n(E) 1

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 If n(E) = 0, then P(E) = 0, and the event is impossible. If E is an event from a sample space S of equally likely outcomes, the probability of event E is: If n(E) = n(S), then P(E) = 1 and the event is certain. Definition: Probability Examples: A 6-sided die is rolled once. The event is impossible. The event is certain. P(10) = = 0 P(n  10) = = 1 P(5) = Note that 0  P(E)  1.

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example 1: Two coins are tossed. What is the probability that at least one head comes up? S = {HH, HT, TH, TT} E = {HH, HT, TH} Example 2: A card is drawn at random from a standard deck of 52 cards. What is the probability the card drawn is a face card? S = all 52 cards in the deckn(S) = 52 E = {  J, J,  J,  J,  Q, Q,  Q,  Q,  K, K,  K,  K} n(E) = 12 Examples: Probability

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Two events A and B are mutually exclusive if they have no outcomes in common, A  B = . Example: When a die is tossed, which events are mutually exclusive? A: getting an even numberC: getting 5 or 6.B: getting an odd number The Venn diagram shows that only A  B = , therefore, only events A and B are mutually exclusive. Definition: Mutually Exclusive Events C A 2 4 6 B 1 3 5

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 JJ J JJ B KK QQ A JJ A  B If A and B are events, their union A  B, is the event “A or B” consisting of all outcomes in A or in B or in both A and B. A  B = {  J, J,  J,  J,  Q,  K } Example: A card is drawn at random from a standard deck of 52 cards. A: getting a club face cardB: getting a jack. List the outcomes for the event of getting a club face card or getting a jack. Example: Union of Two Events

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 If A and B are events, their intersection, written A  B, is the event “A and B” consisting of all outcomes common to both A and B. Example: A card is drawn at random from a standard deck of 52 cards. A  B = {  J} A: getting a club face card List the outcomes for the event of getting a club face card and getting a jack. B: getting a jack. JJ KK QQ A JJ J JJ B A  B Example: Intersection of Two Events

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 (T, H) (H, T) S (T, T) (H, H) A (T, H) (H, T) If A is an event, the complement of A, written A, is the event “not A” consisting of all outcomes not in A. Examples: Two coins are flipped. = {(H, H), (T, T)} A List the outcomes for the event not getting one head and one tail? A Definition: Complementary Events Event A is getting one head and one tail.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 B Probability of Union of Two Events If A and B are events, the probability of “A or B” is: A – n( A  B) + A  B = ( + ) + ( + )– n(A  B) = n( A) + n(B)

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Probability of Union of Mutually Exclusive Events If A and B are mutually exclusive, then AB n(A  B) + A and B are mutually exclusive A  B = 0 = n(A) + n(B) = +

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Probability of the Intersection of Two Events Example: A card is drawn at random from a standard deck of 52 cards. What is the probability the card is a red or a queen? “queen” QQ QQ 7 “red” JJ K Q 5 J 9 8 4 6 10 QQ 66 77 22 AA 99 KK 44  10 33 55 88 2 3

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example 2: A card is drawn at random from a standard deck of 52 cards.What is the probability the card is a spade or a club? Since these events are mutually exclusive, Example: Probability of Mutually Exclusive Events P(club or spade) = P(club) + P(spade) =.  J J  6 6  7 7 22  A A  9 9  K K  4 4  10  3 3  5 5 88 QQ “spade”“club” QQ  J J  6 6  7 7  2 2  A A  9 9  K K  4 4  10  3 3  5 5  8 8

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 For example, when flipping two coins, the events “the first coin comes up heads” and “the second coin comes up tails” are independent. Two events are independent if the fact that one event has occurred has no effect on likelihood of the other event. Definition: Independent Events If A and B are independent events, the probability of “A and B” is:

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Example: A card is drawn at random from a standard deck of 52 cards. What is the probability the card is a red queen? B: the card is a queen Events A and B are independent. A: the card is red Example: Independent Events

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Example: A die is tossed. What is the probability of getting 2 or higher? Example: Probability of Complementary Events It is easier to work with the complementary event “getting a 1”which has probability. If A is an event, the probability of the event “not A” is: