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Review Chapter 9: Probability
9.1 Understanding Probability
9.1.1. Defining Probability
9.1.1.1. Probability: the use of numbers to describe the chance of
something happening
9.1.1.2. experiment: observable situation
9.1.1.3. outcome: the result of performing an experiment
9.1.1.4. sample space: list of all possible outcomes from an experiment
 Examples:
o Experiment: Fair coin toss
 Possible outcomes: Heads (H) or Tails (T)
 Sample space: {H, T}
o Experiment: Fair die tossed
 Possible outcomes: Top side of the die shows 1, 2, 3, 4, 5,
or 6 dots
 Sample space: {1, 2, 3, 4, 5, 6}
9.1.1.5. uniform sample space: all outcomes are equally likely
9.1.1.6. event: subset of the sample space
 Examples:
o S = {HH, HT, TH, TT}; event that at least one coin shows a head: E
= {HH, HT, TH}
o S = {0, 1, 2, 3, 4, 5}; event that the horse race winner is less than 3:
E = {0, 1, 2}
Number of elements of E n(E)
9.1.1.7. P(E) 

Number of elements of S n(S)
9.1.2. Identifying Sample Spaces
9.1.2.1. Some basic properties
9.1.2.1.1. The probability of a certain event is 1: P(E) = 1
9.1.2.1.2. The probability of an impossible event is 0: P(E) = 0
9.1.2.1.3. The probability of an event that is NOT certain or impossible is a
real number between 0 and 1, inclusive: 0  P(E)  1
9.1.2.2. Complementary events
9.1.2.2.1. Events A and B, which share no common outcomes but whose
outcomes account for the total sample space, are known as
complementary events, and their probabilities are related
9.1.2.2.1.1.
P(A) + P(B) = 1
9.1.2.2.1.2.
P(A) = 1 – P(B)
9.1.2.2.1.3.
P(B) = 1 – P(A)
9.1.3. The addition property of probability
9.1.3.1. Dealing with mutually exclusive events
9.1.3.1.1. If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)
9.1.3.1.2. P(AB) = P(A) + P(B)
9.1.3.1.3. see Venn diagram fig. 9.2 p. 459
9.1.3.2. Dealing with events that are NOT mutually exclusive
9.1.3.2.1. If A and B are non-mutually exclusive events, then P(A or B) =
P(A) + P(B) – P(A and B)
9.1.3.2.2. P(AB) = P(A) + P(B) – P(AB)
9.2. Connecting Probability to Models and Counting
9.2.1. Representing and Counting Experimental Outcomes
9.2.1.1. Using a tree diagram
9.2.1.1.1. Favorite of many students
9.2.1.1.2. Only works for small, manageable amounts of outcomes
9.2.1.1.3. See fig. 9.4 and 9.5 p. 465
9.2.1.2. Using a box array
9.2.1.2.1. another way to display the sample space
9.2.1.2.2. see fig. 9.6 p. 465
9.2.1.3. Using the multiplication property of outcomes
9.2.1.3.1. If event A has m outcomes and event B has n outcomes, then
the experiment that has event A followed by event B has m x n
outcomes. This property can be generalized to more than 2 events
9.2.2. Probabilities of Independent Events
9.2.2.1. the outcome of one event has no influence on the outcome of a
second event
9.2.2.2. interested in the events happening simultaneously
9.2.2.3. Multiplication property of probabilities involving independent events:
If events A and B are independent, then P(A and B) = P(A) x P(B)
9.2.2.4. P(AB) = P(A) x P(B)
9.2.3. Conditional Probability
9.2.3.1. Probability of one event is affected by a previous event or
knowledge of a previous event
9.2.3.2. P(A|B) is read the probability of A given B
9.4. Odds and Long Term Behavior
9.4.1. Odds in Favor and Odds Against
P( A )
9.4.1.1. Odds in favor of A =
1  P( A )
1 P( A )
9.4.1.2. Odds against A =
P( A )
9.4.1.3. Alternate way of thinking about this:
9.4.1.3.1. let m be the odds in favor and n be the odds against something
9.4.1.3.2. odds in favor m:n
9.4.1.3.3. odds against n:m
9.4.1.4. Given odds what was the probability of m or n?
9.4.1.4.1. P(m) = m:(m + n)
9.4.1.4.2. P(n) = n:(m + n)
9.4.2. Expected Value
9.4.2.1. average payoff from a game after repeated plays
9.4.2.2. EV = a1P1 + a2P2 + a3P3 + … +anPn
9.4.3. Connecting Expected Value and Odds to Fairness
9.4.3.1. unfair game: EV of payoff  cost of the game
9.4.3.2. Fair game: EV of payoff = cost of the game
9.4.3.3. Expectation of winning = cost of game
9.5. Permutations and Combinations
9.5.1. Listing Permutations
9.5.1.1. Permutations
9.5.1.1.1. A permutation of a set of elements is an arrangement of the
elements in a specified order
9.5.1.1.2. Order IS important
9.5.1.2. Finding the Number of Possible Permutations for a Set of Objects
9.5.1.2.1. How many different ways can a set be ordered?
9.5.1.2.2. How many different orders can a set be placed in?
9.5.1.2.3. If nPn represents the number of permutations of n objects, taken
all together, nPn = n!
9.5.1.2.4. TI-83+ method one – works for all permutation problems
9.5.1.2.4.1.
enter the number for n
9.5.1.2.4.2.
MATH
9.5.1.2.4.3.
PRB
9.5.1.2.4.4.
2: nPr
9.5.1.2.4.5.
enter the number for n
9.5.1.2.4.6.
ENTER
+
9.5.1.2.5. TI-83 method two – works ONLY nPn problems
9.5.1.2.5.1.
enter the number for n
9.5.1.2.5.2.
MATH
9.5.1.2.5.3.
PRB
9.5.1.2.5.4.
4: !
9.5.1.2.5.5.
ENTER
9.5.1.3. Permutations where some, but NOT all, of the objects in a set are
used
9.5.1.3.1. If nPr represents the number of permutations of r objects that
n!
can be formed from a set of n objects, then n Pr 
(n  r )!
9.5.2. Solving Permutation Problems
9.5.2.1. Look over the example on pp.491-2 carefully
9.5.3. Counting Combinations
9.5.3.1. A combination is a subset of a set of elements selected without
regard to their order
9.5.3.2. Order is NOT important
9.5.3.3. If nCr represents the number of combinations of r objects that can
P
n!
be selected from a set of n objects, then n C r  n r 
r!
r! (n  r )!
+
9.5.3.4. TI-83
9.5.3.4.1.1.
enter the number for n
9.5.3.4.1.2.
MATH
9.5.3.4.1.3.
PRB
9.5.3.4.1.4.
3: nCr
9.5.3.4.1.5.
enter the number for n
9.5.3.4.1.6.
ENTER
Chapter Summary: p. 506-507
Chapter Key Terms, Concepts, and Generalizations: p. 507-508
Chapter Review: p. 508-509