Probability of Independent and Dependent Events and Review

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Probability of Independent and Dependent Events and Review. Probability &amp; Statistics

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### Probability of Independent and Dependent Events and Review

Probability & Statistics

1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.

2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

Probability of Independent and Dependent Events and Review

Objectives

Key Words

Independent Events

The occurrence of one event does not affect the occurrence of the other

Dependents Events

The occurrence of one event doesaffect the occurrence of the other

Conditional Probability

Two dependent events A and B, the probability that B will occur given that A has occurred.

• Solve for the probability of an independent event.
• Solve for the probability of a dependent event.

Example 1

Tell whether the events are independent or dependent. Explain.

a.

Your teacher chooses students at random to present their projects. She chooses you first, and then chooses Kim from the remaining students.

b.

You flip a coin, and it shows heads. You flip the coin again, and it shows tails.

c.

One out of 25 of a model of digital camera has some random defect. You and a friend each buy one of the cameras. You each receive a defective camera.

Identify Events

Example 1

SOLUTION

a.

Dependent; after you are chosen, there is one fewer student from which to make the second choice.

b.

Independent; what happens on the first flip has no effect on the second flip.

c.

Independent; because the defects are random, whether one of you receives a defective camera has no effect on whether the other person does too.

Identify Events

Checkpoint

1.

Tell whether the events are independent or dependent. Explain.

You choose Alberto to be your lab partner. Then Tia chooses Shelby.

dependent

2.

You spin a spinner for a board game, and then you roll a die.

independent

Identify Events

Example 2

Concerts A high school has a total of 850 students. The table shows the numbers of students by grade at the school who attended a concert.

a.

What is the probability that a student at the school attended the concert?

b.

What is the probability that a junior did not attend the concert?

Find Conditional Probabilities

Did not attend

Attended

Freshman

80

120

Sophomore

132

86

Junior

173

29

Senior

179

51

Example 2

juniors who did not attend

b.

P(did not attend junior)

=

total juniors

29

0.144

=

=

173

29

+

~

~

~

~

SOLUTION

total who attended

80

132

173

179

+

+

+

a.

P(attended)

=

=

850

total students

564

29

0.664

=

850

202

Find Conditional Probabilities

Checkpoint

3.

Use the table below to find the probability that a student is a junior given that the student did not attend the concert.

Did not attend

0.101

Attended

Freshman

80

120

Sophomore

132

86

Junior

173

29

~

~

Senior

179

51

29

286

Find Conditional Probabilities

Probability of Independent and Dependent Events
• Independent Events
• If A and B are independent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B)
• Dependent Events
• If A and B are dependent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B|A)

Example 3

Independent and Dependent Events

Games A word game has 100 tiles, 98 of which are letters

and two of which are blank. The numbers of tiles of each letter are shown in the diagram. Suppose you draw two

tiles. Find the probability that both tiles are vowels in the situation described.

a.

You replace the first tile before drawing the second tile.

b.

You do not replace the first tile

before drawing the second tile.

Example 3

42

42

(

(

(

A and B

P

(

P

A

(

P

B

(

0.1764

=

=

=

100

100

Independent and Dependent Events

SOLUTION

a.

If you replace the first tile before selecting the second, the events are independent. Let A represent the first tile being a vowel and B represent the second tile

being a vowel. Of 100 tiles,

9

12

9

8

4

42

+

+

+

+

=

are vowels.

Example 3

b.

If you do not replace the first tile before selecting the second, the events are dependent. After removing the first vowel, 41 vowels remain out of 99 tiles.

42

41

~

(

(

~

|

P

A

(

P

B

A

(

0.1739

=

=

100

99

(

A and B

P

(

Independent and Dependent Events

Checkpoint

1

0.0001;

=

10,000

1

~

~

0.0002

4950

Find Probabilities of Independent and Dependent Events

4.

In the game in Example 3, you draw two tiles. What is the probability that you draw a Q, then draw a Z if you first replace the Q? What is the probability that you draw both of the blank tiles (without replacement)?

Conclusion

Summary

Assignment

Probability of Independent and Dependent Events

Page 572

#(11-14,15,18,22,26,30)

• How are probabilities calculated for two events when the outcome of the first event influences the outcome of the second event?
• Multiply the probability of the second event, given that the first event happen.

### Review

Probability & Statistics

1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.

2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

Number of outcomes in event

3

P(red)

=

=

Total number of outcomes

8

Theoretical Probability of an Event

When all outcomes are equally likely, the theoretical probability that an event A will occur is:

The theoretical probability of an event is often simply called its probability.

Example:

What is the probability that the spinner shown lands on red if it is equally likely to land on any section?

Solution:

The 8 sections represent the 8 possible outcomes. Three outcomes correspond to the event “lands on red.”

Number preferring sneakers

P(prefers sneakers)

=

Total number of students

820

0.48

=

1700

~

~

~

~

Number preferring shoes or boots

P(prefers shoes or boots)

=

Total number of students

340

0.19

=

1700

Experimental Probability of an Event

For a given number of trials of an experiment, the experimental probability that an event A will occur is:

Solution:

Find the total number of students surveyed.

820+556+204+120=1700

a. Of 1700 students, 820 prefer sneakers.

b. Of 1700 students surveyed, prefer shoes or boots.

Example:

SurveysThe graph shows results of a survey asking students to name their favorite type of footwear. What is the experimental probability that a randomly chosen student prefers

Sneakers?

Shoes or boots?

Probability of Compound Events
• Overlapping Events
• If A and B are overlapping events, then P(A and B)≠0, and the probability of A or B is:
• Disjoint Events
• If A and B are disjoint events, then P(A and B)=0, and the probability of A or B is:
Probability of the Complement of an Event

The sum of the probabilities of an event and its complement is 1.

So,

Recall:

Complement of an Event

All outcomes that are not in the event

Probability of Independent and Dependent Events
• Independent Events
• If A and B are independent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B)
• Dependent Events
• If A and B are dependent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B|A)