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IntroductionIn probability, events are either dependent or independent. Two events are independent if the occurrence or non-occurrence of one event has no effect on the probability of the other event. If two events are independent, then you can simply multiply their individual probabilities to find the probability that both events will occur. If events are dependent, then the outcome of one event affects the outcome of another event. So it is important to know whether or not two events are independent.
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7.1.3: Understanding Independent Events
Introduction, continuedTwo events A and B are independent if and only if P(A and B) = P(A) • P(B). But sometimes you only know two of these three probabilities and you want to find the third. In such cases, you can’t test for independence, so you might assess the situation or nature of the experiment and then make an assumption about whether or not the events are independent. Then, based on your assumption, you can find the third probability in the equation.
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7.1.3: Understanding Independent Events
Key Concepts• Two events A and B are independent if and only if
they satisfy the following test:
P(A and B) = P(A) • P(B) • Using set notation, the test is• Sometimes it is useful or necessary to make an
assumption about whether or not events are independent, based on the situation or nature of the experiment. Then any conclusions or answers are based on the assumption.
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7.1.3: Understanding Independent Events
Key Concepts, continued• In a uniform probability model, all the outcomes of an
experiment are assumed to be equally likely, and the probability of an event E, denoted P(E), is given by
• When this definition of probability is used and the relevant probabilities are known, then the definition of independent events can be used to determine, verify, or prove that two events are dependent or independent.
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7.1.3: Understanding Independent Events
Key Concepts, continued• The relative frequency of an event is the number of
times it occurs divided by the number of times the experiment is performed (called trials) or the number of observations:
• Relative frequency can be used to estimate probability in some cases where a uniform probability model does not seem appropriate. Such cases include data collected by surveys or obtained by observations.
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7.1.3: Understanding Independent Events
Key Concepts, continued• When relative frequency is used to estimate the
relevant probabilities, then the definition of independent events can be used to determine whether two events seem to be dependent or independent, based on the data.
• Probability and relative frequency are related as follows:
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7.1.3: Understanding Independent Events
Key Concepts, continued
• The probability of an event can be used to predict
its relative frequency if the experiment is performed
a large number of times. For example, the
probability of getting a 3 by rolling a fair die is .
So if you roll a fair die 6,000 times, it is reasonable
to predict that you will get a 3 about 1,000 times, or
of the number of times you roll the die. If you roll
a die 6,000 times, the number of times you get a 3
might not be 1,000.
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7.1.3: Understanding Independent Events
Key Concepts, continued
• Relative frequency can be used to predict the
probability of an event. In general, as the number
of trials or observations increases, the prediction
becomes stronger. For example, suppose you ask
40 people for their favorite ice cream flavor. If 8 say
chocolate, then you can predict that the probability
of a randomly selected person saying chocolate is
, or 20%. But 40 people make up a small
sample, so this is not a very strong prediction. 8
7.1.3: Understanding Independent Events
Key Concepts, continued
Now suppose you ask 4,000 people who are
randomly selected using a good sampling method.
If 740 say chocolate, then you can predict that the
probability of a randomly selected person saying
chocolate is , or 18.5%; this is a stronger
prediction.
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7.1.3: Understanding Independent Events
Key Concepts, continued• Also remember the Addition Rule: If A and B are
any two events, then the probability of A or B, denoted P(A or B), is given byP(A or B) = P(A) + P(B) – P(A and B).
• Using set notation, the rule is
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7.1.3: Understanding Independent Events
Common Errors/Misconceptions• thinking that the actual relative frequency of an event
will equal the probability of the event; for example, thinking 100 tosses of a fair coin will yield 50 heads and 50 tails
• thinking that a probability based on actual relative frequency is a “true fact;” that is, not understanding that the probability is just a number that is only as good as the assumptions and statistical sampling methods it is based on
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7.1.3: Understanding Independent Events
Common Errors/Misconceptions, continued• thinking that a probability based on past actual relative
frequency and obtained using reasonable assumptions and sound statistical sampling methods can be used to guarantee the actual future relative frequency of an event; that is, not realizing that even a probability obtained by the best possible means is only a predictor, never a guarantee
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7.1.3: Understanding Independent Events
Guided Practice
Example 2Trevor tosses a coin 3 times. Consider the following events.
For each of the following pairs of events, determine if the events are independent.
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
1. List the sample space. Sample space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
2. Use the sample space to determine the relevant probabilities.
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7.1.3: Understanding Independent Events
There are 4 outcomes with heads first.
There are 4 outcomes with heads second.
There are 2 outcomes with exactly 2 consecutive heads.
Guided Practice: Example 2, continued
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7.1.3: Understanding Independent Events
There are 2 outcomes with heads first and heads second.
There is 1 outcome with heads first and exactly 2 consecutive heads.
There are 2 outcomes with heads second and exactly 2 consecutive heads.
Guided Practice: Example 2, continued
3. Use the definition of independence to determine if the events are independent in each specified pair.
A and B are independent.
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7.1.3: Understanding Independent Events
Guided Practice: Example 2, continued
A and C are independent.
B and C are dependent.
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7.1.3: Understanding Independent Events
✔
Guided Practice: Example 2, continued
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7.1.3: Understanding Independent Events
Guided Practice
Example 3Landen owns a delicatessen. He collected data on sales of his most popular sandwiches for one week and recorded it in the table below.
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7.1.3: Understanding Independent Events
Breadchoice
Sandwich choice
Landen’s club
Turkey melt
Roasted chicken
Veggie delight
Country white 44 25 25 8
Whole wheat 24 28 26 34
Sourdough 24 27 24 31
Guided Practice
Example 3, continuedEach of the following statements describes a pair of events. For each statement, determine if the events seem to be independent based on the data in the table.
A random customer orders Landen’s club sandwich on country white bread.
A random customer orders the roasted chicken sandwich on whole wheat bread.
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
1. Find the totals of all the categories.
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7.1.3: Understanding Independent Events
Breadchoice
Sandwich choiceTotalLanden’s
clubTurkey
meltRoasted chicken
Veggie delight
Country white 44 25 25 8 102Whole wheat 24 28 26 34 112Sourdough 24 27 24 31 106
Total 92 80 75 73 320
Guided Practice: Example 3, continued
2. For the first statement, assign variables as names of the events. LC: A random customer orders Landen’s club sandwich.
CW: A random customer orders country white bread.
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
3. Use the data to determine the relevant probabilities.
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7.1.3: Understanding Independent Events
There were 320 sandwiches sold, and 92 of them were Landen’s club.
There were 320 sandwiches sold, and 102 of them were on country white bread.
There were 320 sandwiches sold, and 44 of them were Landen’s club on country white bread.
Guided Practice: Example 3, continued
4. Use the definition of independence to determine if the events seem to be independent.
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7.1.3: Understanding Independent Events
Substitute probabilitiesand simplify.
LC and CW seem to be dependent, based on the data.
Guided Practice: Example 3, continued
5. For the second statement, assign variables as names of the events. RC: A random customer orders a roasted chicken sandwich.
WW: A random customer orders whole wheat bread.
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7.1.3: Understanding Independent Events
Guided Practice: Example 3, continued
6. Use the data to determine the relevant probabilities.
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7.1.3: Understanding Independent Events
There were 320 sandwiches sold, and 75 of them were roasted chicken.
There were 320 sandwiches sold, and 112 of them were on whole wheat bread.
There were 320 sandwiches sold, and 26 of them were roasted chicken on whole wheat bread.
Guided Practice: Example 3, continued
7. Use the definition of independence to determine if the events seem to be independent.
RC and WW seem to be independent
based on the data.28
7.1.3: Understanding Independent Events
Substitute probabilities and simplify.
✔
Guided Practice: Example 3, continued
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7.1.3: Understanding Independent Events
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