Introduction In probability, events are either dependent or independent. Two events are independent if the occurrence or non-occurrence of one event has

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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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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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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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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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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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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• 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



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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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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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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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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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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Guided Practice: Example 2, continued

1. List the sample space. Sample space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

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2. Use the sample space to determine the relevant probabilities.

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There are 4 outcomes with heads first.

There are 4 outcomes with heads second.

There are 2 outcomes with exactly 2 consecutive heads.


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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.


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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A and C are independent.

B and C are dependent.

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✔


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http://www.walch.com/ei/00293

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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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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1. Find the totals of all the categories.

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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


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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3. Use the data to determine the relevant probabilities.

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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.


4. Use the definition of independence to determine if the events seem to be independent.

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Substitute probabilitiesand simplify.

LC and CW seem to be dependent, based on the data.


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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6. Use the data to determine the relevant probabilities.

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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.


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


Substitute probabilities and simplify.

✔


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Documents

Introduction In probability, events are either dependent or independent. Two events are independent if the occurrence or non-occurrence of one event has