Digital Lesson Probability

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. The number of outcomes in the sample space S is n(S). Examples: Identify the sample space. Experiment Sample Space n(S) Flip a coin. S = {H, T} 2 Toss a die. S = {1, 2, 3, 4, 5, 6} 6 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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

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

Examples: Probability 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 deck n(S) = 52 E = {J, J, J, J, Q, Q, Q, Q, K, K, K, K} n(E) = 12 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Examples: Probability

Definition: Mutually Exclusive Events 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 number B: getting an odd number C: getting 5 or 6. A 2 4 6 B 1 3 5 C The Venn diagram shows that only A  B = , therefore, only events A and B are mutually exclusive. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Mutually Exclusive Events

Example: Union of Two Events 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. Example: A card is drawn at random from a standard deck of 52 cards. A: getting a club face card B: getting a jack. K Q A J J J J B A  B List the outcomes for the event of getting a club face card or getting a jack. A  B = {J, J, J, J, Q, K } Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Union of Two Events

Example: Intersection of Two Events 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: getting a club face card B: getting a jack. J K Q A J J J B A  B List the outcomes for the event of getting a club face card and getting a jack. A  B = {J} Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Intersection of Two Events

Definition: Complementary Events 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. Event A is getting one head and one tail. (T, H) (H, T) S (T, T) (H, H) A (T, H) (H, T) A List the outcomes for the event not getting one head and one tail? = {(H, H), (T, T)} A Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Complementary Events

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

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

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

Example: Probability of Mutually Exclusive Events 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? “spade”  J  6  7 2  A  9  K  4  10  3  5 8 Q Q  J  6  7  2  A  9  K  4  10  3  5  8 “club” Since these events are mutually exclusive, P(club or spade) = P(club) + P(spade) = . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Probability of Mutually Exclusive Events

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

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

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