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HAWKES LEARNING SYSTEMS
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Section 4.2
Probability Rules
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HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Facts about Probability:
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
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
The Complement:
The complement for E, denoted Ec, consists
of all outcomes in the sample space that are
not in E.
Probability Rule for the Complement:
For an event E and its complement Ec:
P(E) = 1 – P(Ec)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Describe the complement for each of the following events:
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.
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Find the probability:
Roll a pair of dice. What is the probability that neither die is
a 3?
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.
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Addition Rule:
The probability that one event happens or the
other event happens.
Addition Rule for Probability:
For two events E and F:
P(E or F) = P(E) + P(F) – P(E and F)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Calculate the probability:
Find the probability of choosing either a spade or a
face card (king, queen, jack) out of a standard deck of
cards.
Solution:
P(E or F) = P(E) + P(F) – P(E and F)
P(spade or face card) = P(spade) + P(face card) – P( spade and face card)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Mutually Exclusive Events:
• Events that share no outcome.
Addition Rule for Mutually Exclusive Events:
If two events, E and F, are mutually exclusive then:
P(E or F) = P(E) + P(F)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Calculate the probability:
What is the probability of drawing a face card
or a seven from a standard deck of cards?
Solution:
P(E or F) = P(E) + P(F)
P(face card or 7) = P(face card) + P(7)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Definitions:
•
•
•
•
•
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.
Multiplication Rule for Probability:
For two independent events, E and F:
P(E and F) = P(E)P(F)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Calculate the probability:
Choose two cards from a standard deck, with
replacement. What is the probability of
choosing a king and then a queen?
Solution:
P(E and F) = P(E)P(F)
P(king and queen) = P(king) P(queen)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Conditional Probability:
• 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”.
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Calculate the 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.
Solution:
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.
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Multiplication Rule for Dependent Events:
For two dependent events, E and F:
P(E and F) = P(E)P(F|E)
HAWKES LEARNING SYSTEMS
Probability, Randomness, and Uncertainty
math courseware specialists
4.2 Probability Rules
Calculate the probability:
What is the probability of choosing two face
cards in a row without replacement?
Solution:
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)