chapter probability 9 9 copyright © 2013, 2010, and 2007, pearson education, inc
TRANSCRIPT
Chapter
Probability
99
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
9-1 How Probabilities are Determined
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
e. the event E that a number both even and prime is drawn
D = {2, 3, 5, 7, 11, 13, 17, 19, 23}, so n(D) = 9.
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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