2010-2011 Workshop Series for YISD Teachers of 6th Grade Math

Conceptual Understanding & Mathematical Thinking

Workshop 4: Sample Space and Probability(Feb 28, 2011)

Kien Lim

Dept. of Mathematical Sciences, UTEP

Item 0

In how many ways can you form a 3-digit even number with the second digit being a prime number?

Note: 058 is considered a 2-digit even number.

What key ideas should student learn in order to truly understand fundamental counting principle?

• Multiplication as systematic exhaustion without duplication (counting all without double counting)

• Representational tools to generate all cases systematicallyo A tree diagramo An exhaustive list of outcomes

What is the fundamental counting principle?

How can we create learning opportunity for students to experience the need for the fundamental counting principle?

Talk To Your Partners

Consider this problem:Bobbie Bear is planning a vacation. With 3 colored shirts and 2 colored pants, how many outfits can he make?

How can we create learning opportunity for students to experience the need for the fundamental counting principle?

Consider this problem:Bobbie Bear is planning a vacation. With 3 colored shirts and 2 colored pants, how many outfits can he make?

Use the Manipulatives to

Create as Many Outfits as

You Possibly Can

http://illuminations.nctm.org/ActivityDetail.aspx?ID=3

Consider this problem:Bobbie Bear is planning a vacation. With 3 colored shirts and 2 colored pants, how many outfits can he make?

How can we create learning opportunity for students to experience the need for the fundamental counting principle?

• Concrete manipulatives to pictorial representation to formal rule

• Ample time to explore

• Opportunity to experience inefficiency (students appreciate the efficiency of a conceptual tool only when they experience the “hardship” without it)

Logging In Procedure

1. Turn-on your clicker

2. Wait until it says “Enter Student ID”(Enter your 5-digit ID)

3. The screen should display “ANS”

Item 1

Consider a standard 52-card deck, with four suits (♥, ♦, ♠, ♣), 13 cards per suit (2-10, J, Q, K, A). Define an event space on the standard deck such that it consists of 52 simple outcomes, one for each card in the deck.

Which of the following is a true statement?

(A) {Black} is not an event.(B) {Black} is an event with 1 simple outcome.(C) {Black} is an event with 26 simple outcomes.(D) {Black} is an event with 52 simple outcomes.(E) None of the above is true.

What is the difference among• An outcome• An event• A sample space

is the result of an action.

is an outcome or a group of outcomes.

is the set of all possible outcomes.

The sample space has 52 simple outcomes.

{Black} is an event with 26 simple outcomes.

Item 2

Consider a standard 52-card deck, with four suits (♥, ♦, ♠, ♣), 13 cards per suit (2-10, J, Q, K, A). Define an event space on the standard deck such that it consists of two outcomes: Black and Red.

Which of the following is a true statement?

(A) {Black} is not an event.(B) {Black} is an event with 1 simple outcome.(C) {Black} is an event with 26 simple outcomes.(D) {Black} is an event with 52 simple outcomes.(E) None of the above is true.

The sample space has 52 simple outcomes.

{Black} is an event with 1 simple outcome.

The event space {Black, Red} has 2 outcomes.

Item 3

Consider a standard 52-card deck, with four suits (♥, ♦, ♠, ♣), 13 cards per suit (2-10, J, Q, K, A). Define an event space on the standard deck such that it contains, as simple outcomes, only the cards that are hearts or diamonds.

Which of the following is a true statement?

(A) {Black} is not an event.(B) {Black} is an event with 1 simple outcome.(C) {Black} is an event with 26 simple outcomes.(D) {Black} is an event with 52 simple outcomes.(E) None of the above is true.

The sample space has 52 simple outcomes.

{Black} is an event with 0 simple outcome.

The event space {Heart, Diamond} has 2 outcomes.

Create probability question involving a bear who has 3 shirts and 2 pants for students to understand the difference among

• An outcome

• An event

• A sample space

Talk To Your Partners

What is the probability that his outfit will have a green shirt?(A) 1/2(B) 1/3(C) 1/4(D) 1/6(E) 1/8

Item 4

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will put them back and randomly choose again.

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will put them back and randomly choose again.

Sample spaceSample space consists of ___ possible simple outcomes.

Prob(green shirt) =

Number of favorable outcomes Number of possible outcomes

6

2Favorable event (green shirt) consists of ___ simple outcomes.

= 26

= 13

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants.

What is the probability that his outfit will have a green shirt?(A) 1/2(B) 1/3(C) 1/4(D) 1/6(E) 1/8

Item 5

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants.

Prob(green shirt) =

Number of favorable outcomes Number of possible outcomes

= 26

= 13

Is this correct?

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants.

Prob(green shirt) =

Is this correct?26

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants.

Prob(green shirt) =

So is 1/4 or is it 1/3?

14

14

14

14

14

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants.

Prob(green shirt) = or

12

12

14

14

14

14

12

12

1

0

1

0

14

13

18

18

18

18

0

14

14

0

Prob(green shirt)

12

14

12

14

= +

18

= + 18

=

14

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt.

What is the probability that his outfit will have a red shirt?(A) 1/2(B) 1/3(C) 1/4(D) 1/6(E) 1/8

Item 6

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt.

Prob(red shirt) = Number of favorable outcomes Number of possible outcomes

Is this correct?= 1

6

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt.

Prob(red shirt) = Number of favorable outcomes Number of possible outcomes

= 16

Is this correct?

12

12

14 1

4

14

14

14 1

4

14

14

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt.

Prob(red shirt) =

Is the answer 1/8 or 1/6?

12

12

14 1

4

14

14

14 1

4

14

14

14

12

= 18

But the sum of six 1/8s is only 6/8, not 1!

18

18

18

18

18

18

Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt.

Prob(red shirt) =

12

12

14 1

4

14

14

14 1

4

14

14

14

12

18

18

18

18

18

18

13

13

13

13

13

13

14

12

13

+

124

+ 124

= 18

= 3 + 124

= 16

What have we

learned? The usefulness of a

tree diagram.

Item 7

A fair die is rolled three times and the results are recorded in the order that they appear.

For each roll, the die lands with the number 1, 2, 3, 4, 5, or 6 facing up.

The following outcome is the LEAST LIKELY to occur:(A) 2 4 2 (B) 3 4 5(C) 5 5 5(D) 6 4 1 (E) All of the above sequences are equally likely

Item 8

Two fair dice are rolled and the resulting numbers facing upwards on the dice is summed.

The following sum is the MOST LIKELY to occur:

(A) Sum = 3 (B) Sum = 5 (C) Sum = 6 (D) Sum = 9 (E) All of the above sums are equally likely

(A) Sum = 3 (B) Sum = 5 (C) Sum = 6 (D) Sum = 9 (E) All of the above sums are equally likely

P( Sum = 3 ) =

P( Sum = 5 ) =

P( Sum = 6 ) =

P( Sum = 9 ) =

2 = 36

118

4 = 36

1 9

5 36

4 = 36

1 9

Source: Tami Dashley

Item 9

Suppose a family is randomly selected from among all families with 3 children. What is the probability that the family has exactly one boy? You may assume that P(boy) = P(girl) for each birth. (A) 1/2(B) 1/3(C) 1/6(D) 1/8(E) 3/8

Why Clickers and Voting? Requires students to participate actively

(Cline, Zullo, & Parker, 2006)

Provides immediate feedback

Facilitates class discussion/debate

Creates a fun atmosphere

Cline, K. S. (2006). Classroom voting in mathematics. Mathematics Teacher, 100 (2). pp. 100-104.