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

Section 4.2

Probability Rules

HAWKES LEARNING SYSTEMS

math courseware specialists

Copyright © 2008 by Hawkes Learning

Systems/Quant Systems, Inc.

All rights reserved.

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



Facts about Probability:

The complement for E, denoted Ec, consists of all outcomes in the sample space that are not in E.





The Complement:

For an event E and its complement Ec:

P(E) = 1 – P(Ec)

Probability Rule for the Complement:

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:





Roll a pair of dice. What is the probability that neither die is a 3?

Find the probability:





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.

The probability that one event happens or the other event happens.





Addition Rule:

For two events E and F:

P(E or F) = P(E) + P(F) – P(E and F)

Addition Rule for Probability:

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:





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)

• Events that share no outcome.





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:

What is the probability of drawing a face card or a seven from a standard deck of cards?






P(E or F) P(E) + P(F)

P(face card or 7) P(face card) + P(7)

Solution:

• 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.





Definitions:

For two independent events, E and F:

P(E and F) = P(E)P(F)

Multiplication Rule for Probability:

Choose two cards from a standard deck, with replacement. What is the probability of choosing a king and then a queen?






P(E and F) P(E)P(F)

P(king and queen) P(king) P(queen)

Solution:

• 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”.





Conditional Probability:





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.


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:





For two dependent events, E and F:

P(E and F) = P(E)P(F|E)

Multiplication Rule for Dependent Events:

What is the probability of choosing two face cards in a row without replacement?






P(E and F) P(E)P(F|E)

P(face card and face card) P(1st face card) P(2nd face card| 1st face card)

Solution:

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