Probability of Independent and Dependent Events and Review

Probability & Statistics1.0 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• Solve for the probability of

an independent event.• Solve for the probability of a

dependent event.

Key Words• Independent Events

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

• Dependents Events– The occurrence of one event

does affect the occurrence of the other

• Conditional Probability– Two dependent events A and B,

the probability that B will occur given that A has occurred.

Example 1 Identify Events

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.

Example 1 Identify Events

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.

Checkpoint Identify Events

ANSWER dependent

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

1.

Tell whether the events are independent or dependent. Explain.

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

2.

ANSWER independent

Example 2 Find Conditional Probabilities

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?

Freshman

Sophomore

Junior

Senior

80

132

179

173

Grade Attended Did not attend

120

86

51

29

Example 2 Find Conditional Probabilities

564850

= ~~ 0.664

SOLUTION

a. 80P(attended)total who attended

total students=

850=

++ +132 173 179

b. P(did not attend junior) =juniors who did not attend

total juniors

=29

173 +

29202

= 0.144~~29

Checkpoint

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

3.

ANSWER29

2860.101~~

Find Conditional Probabilities

Freshman

Sophomore

Junior

Senior

80

132

179

173

Grade Attended Did not attend

120

86

51

29

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 tilebefore drawing the second tile.

Example 3 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 42are vowels.

(

(

P = A(

(

P B(

(

P• =42

100

• = 0.1764A and B42

100

Example 3 Independent and Dependent Events

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.

= A(

(

P B(

(

P• =42100

4199

• 0.1739A

~~|(

(

P A and B

CheckpointFind Probabilities of Independent and Dependent Events

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)?

4.

ANSWER1

10,000 = 0.0001;

14950

~~ 0.0002

Conclusion

Summary• 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.

Assignment• Probability of Independent

and Dependent Events– Page 572– #(11-14,15,18,22,26,30)

Review

Probability & Statistics1.0 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.

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.”

P(red) =38

Number of outcomes in eventTotal number of outcomes

=

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=1700a. Of 1700 students, 820 prefer sneakers.

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

Example:

Surveys The 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

(a) Sneakers?

(b) Shoes or boots?

P(prefers sneakers)Number preferring sneakers

Total number of students=

820

1700= ~~ 0.48

P(prefers shoes or boots)Number preferring shoes or boots

Total number of students= =

340

1700~~ 0.19

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)