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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)