1. Section 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

2. 1.0 ≤ P(E) ≤ 1 2. P(S) = 1, where S is the sample space of all possible outcomes 3. P(Ø) = 0, where Ø is the empty set Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists Facts about Probability:

3. The complement for E, denoted E c, consists of all outcomes in the sample space that are not in E. Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists The Complement: For an event E and its complement E c : P(E) = 1 – P(E c ) Probability Rule for the Complement:

4. a.Choose a red card out of a standard deck of cards. All 26 black cards. b.Out of 31 students in your statistics class, 15 are out sick with the flu. The 16 students that are not sick. c.In your area, 91% of phone customers use PhoneSouth. The other 9% of customers in your area who do not use PhoneSouth. Describe the complement for each of the following events: HAWKES LEARNING SYSTEMS math courseware specialists Probability, Randomness, and Uncertainty 4.2 Probability Rules

5. Roll a pair of dice. What is the probability that neither die is a 3? Find the probability: HAWKES LEARNING SYSTEMS math courseware specialists Probability, Randomness, and Uncertainty 4.2 Probability Rules Solution: It would be tedious to write out every combination that does not contain a 3. Using the compliment, we could list the outcomes which either die contains a 3. There are 11 outcomes where at least one of the dice is a 3.

6. The probability that one event happens or the other event happens. Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists Addition Rule: For two events E and F: P(E or F) = P(E) + P(F) – P(E and F) Addition Rule for Probability:

7. Find the probability of choosing either a spade or a face card (king, queen, jack) out of a standard deck of cards. Calculate the probability: HAWKES LEARNING SYSTEMS math courseware specialists Probability, Randomness, and Uncertainty 4.2 Probability Rules P(E or F)  P(E) + P(F) – P(E and F) Solution: P(spade or face card)  P(spade) + P(face card) – P( spade and face card)

8. Events that share no outcome. Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists Mutually Exclusive Events: If two events, E and F, are mutually exclusive then: P(E or F) = P(E) + P(F) Addition Rule for Mutually Exclusive Events:

9. What is the probability of drawing a face card or a seven from a standard deck of cards? Calculate the probability: HAWKES LEARNING SYSTEMS math courseware specialists Probability, Randomness, and Uncertainty 4.2 Probability Rules P(E or F)  P(E) + P(F) P(face card or 7)  P(face card) + P(7) Solution:

10. With repetition – outcomes may be repeated. Without repetition – outcomes may not be repeated. With replacement – objects are placed back into consideration for the following choice. Without replacement – objects are not placed back into consideration for the following choice. Independent events – if one event happening does not influence the probability of the other event happening. Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists Definitions: For two independent events, E and F: P(E and F) = P(E)P(F) Multiplication Rule for Probability:

11. Choose two cards from a standard deck, with replacement. What is the probability of choosing a king and then a queen? Calculate the probability: HAWKES LEARNING SYSTEMS math courseware specialists Probability, Randomness, and Uncertainty 4.2 Probability Rules P(E and F)  P(E)P(F) P(king and queen)  P(king) P(queen) Solution:

12. When two events are not independent, the outcome of one influences the probability of the other. P(F|E) is read as “the probability of event F occurring given event E occurred first”. Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists Conditional Probability:

13. Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists What is the probability of choosing a red card from the deck, given that the first card drawn was a diamond? Assume the cards are chosen without replacement. Calculate the probability: First we need to determine the number of red cards left in the deck. Since a diamond is a red card and has already been chosen there are only 25 red cards left. Solution:

14. Probability, Randomness, and Uncertainty 4.2 Probability Rules HAWKES LEARNING SYSTEMS math courseware specialists For two dependent events, E and F: P(E and F) = P(E)P(F|E) Multiplication Rule for Dependent Events:

15. What is the probability of choosing two face cards in a row without replacement? Calculate the probability: HAWKES LEARNING SYSTEMS math courseware specialists Probability, Randomness, and Uncertainty 4.2 Probability Rules P(E and F)  P(E)P(F|E) P(face card and face card)  P(1 st face card) P(2 nd face card| 1 st face card) Solution: