### General Information

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Data Analysis and Probability

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

### Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Conditional relative frequency
- Event
- Experimental probability
- Frequency table
- Joint frequency
- Joint relative frequency
- Sample space
- Theoretical probability

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7 and grade 8, students began working with theoretical probabilities and comparing them to experimental probability as well as collecting data through simulation. In Mathematics for College Statistics, students apply the addition rule for probability,*P(A or B) = P(A) + P(B) - P(A and B)*, to calculate the probability of event

*A or B*. Students recognize the effect mutually exclusive events have on this rule and be able to interpret the result in terms of the model and its context.

- Students should be able to calculate the union of two events using the formula
*P(A*∪*B) = P(S) + P(B) - P(A*∩*B)*. - The symbol for union, ∪, is read as “or” and the symbol for intersection, ∩, is read as “and.” If students are trying to identify what action to take from a word problem involving probability, “or” should be the key word letting students know to use the addition rule.
- Instruction includes the use of a reference sheet with the formulas for the addition rule for probability, the multiplication rule for probability and the conditional probability formula.
- Students should be able to explain why we need to subtract the intersection when calculating the union.
- Be sure to distinguish independence from mutually exclusive events.?
- In mutually exclusive events,
*P(A*∩*B)*= 0, which means the two events cannot occur at the same time.? - In mutually exclusive events,
*P(A*∪*B) = P(A) + P(B)*. - Instructions includes using the product of probabilities to check independence
(MA.912.DP.4.2). If
*P(A*∩*B) = P(A) x P(B)*, then the events are independent. - Instruction distinguishes independence from mutually exclusive events in order to differentiate different types of joint probabilities.
- Instruction clarifies that mutually exclusive events
*(A*∩*B)*= 0, which means the two events cannot occur at the same time. - Instruction includes the use of models, which may include Venn Diagrams, organized lists, two-way tables or tree diagrams.
- Students are expected to be able to interpret the results in context as well as how it would be represented on a model such as a Venn Diagram.
- In a college statistics class, students will typically work with the addition rule for probability and mutually exclusive events using contingency tables and Venn diagrams.
- Students should be able to work backwards if given the union to find other missing pieces in the addition rule formula.
- Instruction includes the use of technology whether it be a statistical program or graphing calculators.?
- Instruction includes the use of real-world data.

### Common Misconceptions or Errors

- Students may not recognize events as mutually exclusive or non-mutually exclusive.
- Student may have difficulty when pulling information from a Venn Diagram, especially when they have multiple sections that overlap.
- Students may make errors with determining the correct denominator. They may use the total rather than the specified event.
- Students may have misconceptions in their understanding of the “overlap” in compound events.
- Students may have difficulty with algebraic skills if using the addition rule formula to work backwards and solve for an unknown.

### Instructional Tasks

*Instructional Task 1 (MTR.7.1)*

- In a certain school, 15 percent of the students are enrolled in a physics course, 27 percent are
enrolled in a foreign language course, and 30 percent are enrolled in either a physics course
or a foreign language course or both.
- Part A. What is the probability that a student chosen at random from this school will be enrolled in a physics course?
- Part B. What is the probability that a student chosen at random from this school will be enrolled in a foreign language course?
- Part C. What is the probability that a student chosen at random from this school will be enrolled in both a foreign language course and a physics course?
- Part D. Draw a Venn Diagram to model this situation

*Instructional Task 2 (MTR.4.1, MTR.7.1)*

- If a fair single 6-sided die is rolled, answer the following.
- Part A. What is the probability of rolling a number less than 4 or an even number?
- Part B. What is the probability of rolling a number greater than 5 or an odd?
- Part C. Using statistical terminology, explain why you needed to subtract in Part A. but not for Part B.

Instructional Items

Instructional Items

*Instructional Item 1*

- At The Triple Stack cafe, everyone orders either pancakes, waffles, or French toast. Let
*S*= the event that a randomly selected customer puts syrup on their food. Let*F*= the event that a randomly selected customer orders a fruit topping on their food. Suppose that after years of collecting data, The Triple Stack cafe has estimated the following probabilities:

*P(S) = 0.8*

*P(F) = 0.5*

*P(S or F) = 0.9*

- Estimate
*P(S and F)*and interpret this value in the context of the problem.

Instructional Item 2

Instructional Item 2

- Given the two-way table below, find the following probabilities and draw a Venn-diagram to model each situation.

- Part A. Find the probability that a randomly selected person rides the bus or is a girl.
- Part B. Find the probability that a randomly selected person is a boy or does not ride the bus.
- Part C. What are two events from the table that are not mutually exclusive?
- Part D. Why are the events ‘Ride the Bus’ and ‘Do Not Ride the Bus’ mutually exclusive?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*