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Five-Minute Check (over Lesson 13–4) CCSS Then/Now New Vocabulary Example 1: Identify Independent and Dependent Events Key Concept: Probability of Two Independent Events Example 2: Real-World Example: Probability of Independent Events Key Concept: Probability of Two Dependent Events Example 3: Probability of Dependent Events Example 4: Standardized Test Example: Conditional Probability Key Concept: Conditional Probability Lesson Menu

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

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. 0.059 B. 0.25 C. 0.34 D. 0.425 5-Minute Check 2

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

Mathematical Practices 2 Reason abstractly and quantitatively. Content Standards 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.3 Understand the conditional probability of A given B as , and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Mathematical Practices 2 Reason abstractly and quantitatively. 4 Model with mathematics. CCSS

You found simple probabilities. Find probabilities of independent and dependent events Find probabilities of events given the occurrence of other events. Then/Now

conditional probability probability tree compound event independent events dependent events conditional probability probability tree Vocabulary

A. A die is rolled, and then a second die is rolled. Identify Independent and Dependent Events Determine whether the event is independent or dependent. Explain your reasoning. A. A die is rolled, and then a second die is rolled. Answer: The two events are independent because the first roll in no way changes the probability of the second roll. Example 1

Identify Independent and Dependent Events Determine whether the event is independent or dependent. Explain your reasoning. B. A card is selected from a deck of cards and not put back. Then a second card is selected. 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

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

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

Concept

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, they will order a hamburger. If they draw a red slip, they will order pizza. Suppose Michelle draws a slip. Not liking the outcome, she puts it back and draws a second time. What is the probability that on each draw her slip is green? 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 2

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 2

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 2

Concept

Probability of Dependent Events EATING OUT Refer to Example 2. Recall that there were 5 green slips of paper and 6 red slips of paper in a bag. Suppose that Michelle draws a slip and does not put it back. Then her friend Christina draws a slip. What is the probability that both friends draw a green slip? These events are dependent since Michelle does not replace the slip she removed. Let G represent a green slip and R represent a red slip. Example 3

Probability of dependent events After the first green slip is chosen, 10 total slips remain, and 4 of those are green. Simplify. Answer: So, the probability that both friends draw green slips is or about 18%. Example 3

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 3

Students who draw odd numbers will be on the Red team. Conditional Probability Mr. Monroe is organizing the gym class into two teams for a game. The 20 students randomly draw cards numbered with consecutive integers from 1 to 20. Students who draw odd numbers will be on the Red team. Students who draw even numbers will be on the Blue team. If Monica is on the Blue team, what is the probability that she drew the number 10? Example 4

Let A be the event that an even number is drawn. Conditional Probability Read the Test Item Since Monica is on the Blue team, she must have drawn an even number. So you need to find the probability that the number drawn was 10, given that the number drawn was even. This is a conditional problem. Solve the Test Item Let A be the event that an even number is drawn. Let B be the event that the number drawn is 10. Example 4

Draw a Venn diagram to represent this situation. Conditional Probability Draw a Venn diagram to represent this situation. There are ten even numbers in the sample space, and only one out of these numbers is a 10. Therefore, the P(B | A) = The answer is B. Example 4

Mr. Riley’s class is traveling on a field trip for Science class Mr. Riley’s class is traveling on a field trip for Science class. There are two busses to take the students to a chemical laboratory. To organize the trip, 32 students randomly draw cards numbered with consecutive integers from 1 to 32. Students who draw odd numbers will be on the first bus. Students who draw even numbers will be on the second bus. If Yael will ride the second bus, what is the probability that she drew the number 18 or 22? A. B. C. D. Example 4

Concept

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