# Describing Events

Alignments to Content Standards: S-CP.A.1

The 10 cards below provide information on the ten students on the robotics team at a high school. The students are identified by an ID number: S1, S2, …, S10. For each student, the following information is also given:

Gender

Whether or not the student is currently enrolled in a science class

Whether or not the student is currently participating on a school sports team

Typical number of hours of sleep on a school night

It might be helpful to cut out these cards so that you can sort and rearrange them as you answer the following questions.

1. Suppose that one student on the robotics team will be selected at random to represent the team at an upcoming competition. This can be viewed as a chance experiment. Which one of the following is the sample space for this experiment?

S = { Student ID, gender, grade level, science class, sports team, hours of sleep}
S = {S1, S2, S3, S4, S5, S6, S7, S8, S9, S10}
S = {S1, S2, S3, S4, S5, S6, S7, S8, S9, S10, male, female, 9, 10, 11, 12, yes, no, 6, 7, 8, 9}

2. What outcomes from the sample space are in the event that the selected student is taking a science class?

3. Consider the following three events:
A = the selected student is female
B = the selected students is on a sports team
C = the selected student typically sleeps less than 8 hours on a school night

For each of these three events, list the outcomes that make up the event.

4. Which outcomes are in the following events?
a.A or B
b.A and C
c.not C

5. Describe each of the events in question 4 in words.

6. Define two additional events in the context of this chance experiment. Use the letters D and E to represent these events.

7. For the two events you defined in question 6, describe the event D or E and the event D and E in words.

8. Is the set of outcomes in D or E and the set of outcomes in D and E the same for the two events you have defined? Explain why or why not.

 S1 Gender: Female Grade: 11 Science course:  yes Sports team: no Sleep: 7 hours S2 Gender: Male Grade: 9 Science course:  no Sports team: no Sleep: 9 hours S3 Gender: Male Grade: 11 Science course:  no Sports team: yes Sleep: 8 hours S4 Gender: Male Grade: 10 Science course:  yes Sports team: yes Sleep: 6 hours S5 Gender: Female Grade: 12 Science course:  yes Sports team: yes Sleep: 7 hours S6 Gender: Female Grade: 9 Science course:  yes Sports team: no Sleep: 9 hours S7 Gender: Male Grade: 11 Science course:  yes Sports team: no Sleep: 7 hours S8 Gender: Female Grade: 10 Science course:  no Sports team: no Sleep: 8 hours S9 Gender: Male Grade: 12 Science course:  no Sports team: yes Sleep: 6 hours S10 Gender: Male Grade: 12 Science course:  yes Sports team: no Sleep: 8 hours

## IM Commentary

Probability was first introduced in Grade 7 (7.SP7 and 7.SP8). In high school students extend their understanding of probability, but since it may have been two or more years since students worked with probability, there is some overlap in the grade 7 and the high school probability standards. The purpose of this task is to illustrate standard S.CP.1, which provides a review of the definitions of sample space and events. Students are asked to describe events both verbally and as subsets of a sample space.

Students can work on this task individually or in pairs, but it might be a good idea to talk about question 1 as a class before having them work on the questions that follow. Question 1 will require that students access prior knowledge of the definition of the sample space for a chance experiment. Point out that the chance experiment here is selecting one of the ten students in the robotics club at random, so there are ten possible outcomes. The sample space would be a set corresponding to the 10 students, making the second choice listed the correct choice.

You may also want to point out that when the word “or” is used in defining events, it is not the “exclusive or” that is often used in everyday language.

You may want to print out the page with the student cards and have students cut them apart. This will make it possible for them to divide the cards into piles—this will make it easier to keep track of what outcomes are in various events.

## Solution

1. The correct choice is S = {S1, S2, S3, S4, S5, S6, S7, S8, S9, S10 }. Because the chance experiment consists of selecting one of the 10 students at random, the sample space would consist of 10 outcomes corresponding to the 10 students in the robotics club.

2. S1, S4, S5, S6, S7, S10

3. A: S1, S5, S6, S8
B: S3, S4, S5, S9
C: S1, S4, S5, S7, S9

4. A or B: S1, S3, S4, S5, S6, S8, S9
A and C: S1, S5
not C: S2, S3, S6, S8, S10

5. A or B: the event that the selected student is female or is on a sports team
A and C: the event that the selected student is female and typically sleeps less than 8 hours on a school night
not C: The event that the selected student does not typically sleep less than 8 hours on a school night. The event not C could also be described as the event that the selected student typically sleeps at least 8 hours on a school night.

The answers for questions f – h will vary depending on what events that the student defines. Check to make sure that students have given a correct description of the union and the intersection for the two events that they defined. In question 8, it might be possible for students to define two events for which both the union and the intersection are the same, but this would not usually be the case. Check the explanation carefully if a student answers yes for question h.