1. Five-Minute Check (over Lesson 12–4) Mathematical Practices Then/Now
New Vocabulary
Example 1: Identify Independent and Dependent Events
Example 2: Use Probability to Identify Independent Events
Key Concept: Probability of Independent Events
Example 3: Real-World Example: Probability of Independent Events
Key Concept: Probability of Dependent Events
Example 4: Probability of Dependent Events
Example 5: Real-World Example: Use Probability to Analyze Decisions
Lesson Menu

2. An archer conducts a probability simulation to find that he hits a bull’s eye 21 out of 25 times. What is the probability that he does not hit a bull’s eye?
A. 0.14
B. 0.16
C. 0.18
D. 0.20
5-Minute Check 1

3. The administrators at a high school use a random number generator to simulate the probability of randomly selecting one student. The results are shown in the table. What is the probability of selecting a freshman? A
B. 0.25
C. 0.34 D
5-Minute Check 2

4. A. P(roll a 6): ; roll a number cube 75 times
Which of these experiments is most likely to have results that match the given theoretical probability?
A. P(roll a 6): ; roll a number cube 75 times
B. P(roll a 2 or 3): ; roll a number cube 15 times
C. P(roll an even number ): ; roll a number cube 20 times
D. P(roll a 4): ; roll a number cube 1 time __ 1 6 3 2
5-Minute Check 3

5. Mathematical Practices
1 Make sense of problems and persevere in solving them.
4 Model with mathematics.
Content Standards
S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A) P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
S.MD.7 (+) Analyze decisions and strategies using probability concepts e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). MP

6. You found simple probabilities.
Apply the multiplication rule to situations involving independent events.
Apply the multiplication rule to situations involving dependent events.
Then/Now

7. compound event
independent events
dependent events
Vocabulary

8. A. Anna rolls a 6 on one number cube and a 3 on another cube.
Identify Independent and Dependent Events
Determine whether the event is independent or dependent. Explain your reasoning.
A. Anna rolls a 6 on one number cube and a 3 on another cube.
The outcome of the first roll in no way affects the outcome of the second roll.
This makes the events independent.
Answer: The two events are independent, because the outcome on the first die does not affect the outcome on the second die.
Example 1

9. This makes the two events dependent.
Identify Independent and Dependent Events
Determine whether the event is independent or dependent. Explain your reasoning.
B. A queen is selected from a standard deck of cards and not put back. Then a king is selected.
After a queen is chosen, that card cannot be selected again. This affects the probability that a king is chosen, because the sample space is reduced by one card.
This makes the two events dependent.
Answer: The two events are dependent because the first card is removed and cannot be selected again. This affects the probability of the second draw because the sample space is reduced by one card.
Example 1

10. Determine whether the event is independent or dependent
Determine whether the event is independent or dependent. Explain your reasoning.
A. A marble is selected from a bag. It is not put back. Then a second marble is selected.
A. independent
B. dependent
Example 1

11. Determine whether the event is independent or dependent
Determine whether the event is independent or dependent. Explain your reasoning.
B. A marble is selected from a bag. Then a card is selected from a deck of cards.
A. independent
B. dependent
Example 1

12. Use Probability to Identify Independent Events
A bag contains a white marble, a blue marble, a yellow marble, and a green marble. Andrew selects the white marble, replaces it, and then selects the green marble. Are these events independent? Explain using probability.
Does selected a white marble and returning it change the outcome of the second draw? No
Explain with probability.
P(WG) = P(W) × P(G)
Example 2

13. Are the events independent or dependent?
Use Probability to Identify Independent Events
Are the events independent or dependent?
Independent, because P(WG) = P(W) × P(G).
The product of individual probabilities is equal to the probability of the two events occurring together. Therefore; the two events are independent.
Answer: Yes; the sample space has 16 equally likely outcomes: {WW, WB, WY, WG, BW, BB, BY, BG, YW, YB, YY, YG, GW, GB, GY, GG}.
; therefore, the two events are independent.
Example 2

14. Concept

15. Probability of Independent Events
EATING OUT Michelle and Christina are going out to lunch. They put 5 green slips of paper and 6 red slips of paper into a bag. If a person draws a green slip, she will order a hamburger. If she draws a red slip, she will order pizza. Michelle will draw first and put her slip back. Then Christina will draw. What is the probability that both girls draw green slips?
These events are independent since Michelle replaced the slip that she removed. Let G represent a green slip and R represent a red slip.
Example 3

16. Probability of independent events
Draw 1
Draw 2
Probability of independent events
Answer: So, the probability that on each draw Michelle’s slips were green is
Example 3

17. LABS In Science class, students are drawing marbles out of a bag to determine lab groups. There are 4 red marbles, 6 green marbles, and 5 yellow marbles left in the bag. Jacinda draws a marble, but not liking the outcome, she puts it back and draws a second time. What is the probability that each of her 2 draws gives her a red marble?
A. 12.2%
B. 10.5%
C. 9.3%
D. 7.1%
Example 3

18. Concept

19. The first three players all win prizes. Find each probability.
Probability of Dependent Events
GAMES At the school carnival, winners in the ringtoss game are randomly given a prize from a bag that contains 4 sunglasses, 6 hairbrushes, and 5 key chains.
The first three players all win prizes. Find each probability.
A. P (sunglasses, hairbrush, key chain)
After a prize is won, that prize cannot be won again. This affects the probability of the next prize won.
This makes the three events dependent.
Example 4

20. The probability of winning sunglasses first.
Probability of Dependent Events
P(sunglasses) =
The probability of winning sunglasses first.
P(hairbrush) =
The probability of winning a hairbrush second.
P(key chain) =
The probability of winning a key chain third.
P(sunglasses, hairbrush, key chain) =
Answer: The probability of winning sunglasses then a
hairbrush and then a key chain is
Example 4

21. The first three players all win prizes. Find each probability.
Probability of Dependent Events
GAMES At the school carnival, winners in the ringtoss game are randomly given a prize from a bag that contains 4 sunglasses, 6 hairbrushes, and 5 key chains.
The first three players all win prizes. Find each probability.
B. P (hairbrush, hairbrush, not a hairbrush)
After a prize is won, that prize cannot be won again. This affects the probability of the next prize won.
This makes the three events dependent.
Example 4

22. The probability of winning a hairbrush first.
Probability of Dependent Events
P(hairbrush) =
The probability of winning a hairbrush first.
P(hairbrush) =
The probability of winning a hairbrush second.
P(not a hairbrush) =
The probability of not winning a hair brush third.
P(hairbrush, hairbrush, not a hairbrush) =
Answer: The probability of winning a hairbrush then a
hairbrush and then not a hairbrush is
Example 4

23. LABS In Science class, students are again drawing marbles out of a bag to determine lab groups. There are 4 red marbles, 6 green marbles, and 5 yellow marbles. This time Graham draws a marble and does not put his marble back in the bag. Then his friend Meena draws a marble. What is the probability they both draw green marbles?
A. B.
C. D.
Example 4

24. Find the probability of rain on Saturday and Sunday.
Use Probability to Analyze Decisions
Davina’s family will cancel their weekend camping trip if the probability of rain on both Saturday and Sunday is greater than 10%. According to the weather forecast, there is a 30% chance of rain on Saturday and a 20% chance of rain on Sunday. Assuming the two events (rain on Saturday and rain on Sunday) are independent, should Davina’s family cancel the trip? Justify your answer using probability
Find the probability of rain on Saturday and Sunday.
Example 5

25. P(Sat and Sun) = P(Sat) × P(Sun)
Use Probability to Analyze Decisions
P(Sat and Sun) = P(Sat) × P(Sun)
Should Davina’s family cancel their weekend trip?
No, there is only a 6% chance of rain.
Answer: No,
Example 5