Check it out!

1

1.3.2: Simple Random Sampling

Players of a dice game roll five dice and earn points according to the combinations of numbers they roll. After the first roll, a player can pick up as many dice as desired and re-roll them to get an improved score. The following table outlines the points scored for different types of rolls.

2

1.3.2: Simple Random Sampling

Points Description Example roll0 All five dice have different numbers. 1, 2, 3, 5, 61 Two dice have the same number. 1, 2, 2, 3, 52 Two sets of two dice have the same number. 1, 3, 3, 6, 63 Three dice have the same number. 2, 2, 2, 4, 5

4Three dice have the same number and the other two dice match. 1, 1, 1, 4, 4

5 Four dice have the same number. 3, 3, 3, 3, 66 All five dice have the same number. 5, 5, 5, 5, 5

To better understand the game, Chad simulated rolling dice on his graphing calculator. He played 20 games with the following results. Use the table to answer the questions that follow.

3

1.3.2: Simple Random Sampling

Initial roll Points Second roll Total points

4, 3, 6, 2, 3 1 3, 3, 4, 6, 6 2

6, 1, 3, 6, 2 1 6, 6, 1, 1, 3 2

4, 2, 5, 5, 1 1 5, 5, 4, 1, 5 3

5, 6, 1, 2, 4 0 2, 1, 6, 2, 2 3

1, 2, 1, 2, 5 2 1, 1, 2, 2, 6 2

4, 1, 3, 3, 4 2 3, 3, 4, 4, 6 2

6, 6, 5, 2, 5 2 6, 6, 5, 5, 4 2

1, 2, 4, 3, 1 2 1, 1, 4, 4, 4 4

3, 1, 4, 3, 1 2 3, 3, 1, 1, 5 2

(continued)

4

1.3.2: Simple Random Sampling

Initial roll Points Second roll Total points

6, 2, 6, 5, 1 1 6, 6, 1, 3, 4 1

4, 5, 3, 1, 1 1 1, 1, 5, 5, 1 2

3, 4, 2, 4, 3 2 3, 3, 4, 4, 1 2

1, 3, 2, 3, 1 2 1, 1, 3, 3, 3 4

2, 2, 1, 1, 1 4 2, 2, 1, 1, 1 4

3, 4, 1, 1, 3 2 3, 3, 1, 1, 5 2

6, 3, 1, 5, 5 1 5, 5, 1, 1, 1 4

6, 6, 6, 3, 5 3 6, 6, 6, 4, 6 5

4, 3, 5, 4, 5 2 4, 4, 5, 5, 5 4

2, 2, 4, 5, 4 2 2, 2, 4, 4, 3 2

4, 2, 6, 4, 3 1 4, 4, 3, 4, 4 5

Total 34 Total 57

1. What percent of the time could you expect to earn 0 points on the first roll based on the results of Chad’s simulation?

2. Suppose it is your final turn and you need a minimum of 3 points to win the game. What is the probability that you will win according to the simulation?

3. What is the mean of the values for each of the first rolls in the simulation?

4. What is the mean of the values after each of the second rolls in the simulation?

5

1.3.2: Simple Random Sampling

1. What percent of the time could you expect to earn 0 points on the first roll based on the results of Chad’s simulation?• To determine the percent of the time you would

expect to earn 0 points on the first roll, first find the number of times in the simulation that the first roll yielded 0 points. According to the simulation, only 1 roll earned 0 points.

6

1.3.2: Simple Random Sampling

• Calculate the percent by setting up a ratio of the number of occurrences of 0 points out of the total number of first rolls, 20.

• You could expect to earn 0 points on the first roll 5% of the time, according to the simulation.

7

1.3.2: Simple Random Sampling

2. Suppose it is your final turn and you need a minimum of 3 points to win the game. What is the probability that you will win according to the simulation?• To determine the probability that you will win, find

the number of games in the simulation for which the total points earned for both rolls was at least 3.

• There were 9 games in the simulation with a total score that was greater than or equal to 3 points.

• According to the simulation, the probability that you will win is 45%.

8

1.3.2: Simple Random Sampling

3. What is the mean of the values for each of the first rolls in the simulation?

Recall that the mean of a data set can be found using

the formula

, where n is the number

of data values (x).

9

1.3.2: Simple Random Sampling

Formula for

calculating mean

Substitute the known values. (Repeated values are listed as products for convenience.)

Simplify.

• The mean of the values for the first rolls is 1.7.10

1.3.2: Simple Random Sampling

4. What is the mean of the values after each of the second rolls in the simulation?

Again, use the formula

to calculate the mean.

11

1.3.2: Simple Random Sampling

Formula for calculating mean

Substitute the known values. (Repeated values are listed as products for convenience.)

Simplify.

• The mean of the values after each of the secondrolls is 2.85.

12

1.3.2: Simple Random Sampling