Download 9-1 How Probabilities are Determined

Chapter
9
Probability
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9-1 How Probabilities are Determined
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Determining Probabilities
Mutually Exclusive Events
Complementary Events
Non-Mutually Exclusive Events
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Definitions
Experiment: an activity whose results can be
observed and recorded.
Outcome:
each of the possible results of an
experiment.
Sample space: a set of all possible outcomes for
an experiment.
Event:
any subset of a sample space.
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Example 9-1
Suppose an experiment consists of drawing 1 slip
of paper from a jar containing 12 slips of paper,
each with a different month of the year written on
it. Find each of the following:
a. the sample space S for the experiment
S = {January, February, March, April, May, June,
July, August, September, October, November,
December}
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Example 9-1
(continued)
b. the event A consisting of outcomes having a
month beginning with J
A = {January, June, July}
c. the event B consisting of outcomes having the
name of a month that has exactly four letters
B = {June, July}
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Example 9-1
(continued)
d. the event C consisting of outcomes having a
month that begins with M or N
C = {March, May, November}
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Determining Probabilities
Experimental (empirical) probability:
determined by observing outcomes of
experiments.
Theoretical probability:
the outcome under ideal conditions.
Equally likely:
when one outcome is as likely as another
Uniform sample space:
each possible outcome of the sample space is
equally likely.
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Law of Large Numbers
(Bernoulli’s Theorem)
If an experiment is repeated a large number of
times, the experimental (empirical) probability of a
particular outcome approaches a fixed number as
the number of repetitions increases.
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Probability of an Event with Equally
Likely Outcomes
For an experiment with sample space S with
equally likely outcomes, the probability of an event
A is
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Example 9-2
Let S = {1, 2, 3, 4, 5, …, 25}. If a number is chosen
at random, that is, with the same chance of being
drawn as all other numbers in the set, calculate
each of the following probabilities:
a. the event A that an even number is drawn
A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}, so
n(A) = 12.
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Example 9-2
(continued)
b. the event B that a number less than 10 and
greater than 20 is drawn
c. the event C that a number less than 26 is drawn
C = S, so n(C) = 25.
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Example 9-2
(continued)
d. the event D that a prime number is drawn
D = {2, 3, 5, 7, 11, 13, 17, 19, 23}, so n(D) = 9.
e. the event E that a number both even and prime
is drawn
E = {2}, so n(E) = 1.
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Definitions
Impossible event:
an event with no outcomes; has probability 0.
Certain event: an event with probability 1.
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Probability Theorems
If A is any event and S is the sample space, then
The probability of an event is equal to the sum of
the probabilities of the disjoint outcomes making
up the event.
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Example 9-3
If we draw a card at random from an ordinary deck
of playing cards, what is the probability that
a. the card is an ace?
There are 52 cards in a deck, of which 4 are aces.
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Example 9-3
(continued)
If we draw a card at random from an ordinary deck
of playing cards, what is the probability that
b. the card is an ace or a queen?
There are 52 cards in a deck, of which 4 are aces
and 4 are queens.
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Mutually Exclusive Events
Events A and B are mutually exclusive if they
have no elements in common; that is,
For example, consider one spin
of the wheel.
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9},
A = {0, 1, 2, 3, 4}, and
B = {5, 7}.
If event A occurs, then event B cannot occur.
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Mutually Exclusive Events
If events A and B are mutually exclusive, then
The probability of the union of events such that
any two are mutually exclusive is the sum of the
probabilities of those events.
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Complementary Events
Two mutually exclusive events whose union is the
sample space are complementary events.
For example, consider the event A = {2, 4} of
tossing a 2 or a 4 using a standard die. The
complement of A is the set A = {1, 3, 5, 6}.
Because the sample space is S = {1, 2, 3, 4, 5, 6},
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Complementary Events
If A is an event and A is its complement, then
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Non-Mutually Exclusive Events
Let E be the event of spinning
an even number.
E = {2, 14, 18}
Let T be the event of spinning a
multiple of 7.
T = {7, 14, 21}
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Summary of Probability Properties
1. P(Ø) = 0 (impossible event)
2. P(S) = 1, where S is the sample space (certain
event).
3. For any event A, 0 ≤ P(A) ≤ 1.
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Summary of Probability Properties
4. If A and B are events and A ∩ B = Ø, then
P(A U B) = P(A) + P(B).
5. If A and B are any events, then
P(A U B) = P(A) + P(B) − P(A ∩ B).
6. If A is an event, then P(A) = 1 − P(A).
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Example 9-4
A golf bag contains 2 red tees, 4 blue tees, and 5
white tees.
a. What is the probability of the event R that a tee
drawn at random is red?
Because the bag contains a total of 2 + 4 + 5 = 11
tees, and 2 tees are red,
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Example 9-4
(continued)
b. What is the probability of the event “not R”; that
is a tee drawn at random is not red?
c. What is the probability of the event that a tee
drawn at random is either red (R) or blue (B);
that is, P(R U B)?
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