Three dice game I

Three dice game I

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Further Resources
This task is about estimating the probability of throwing a pair.
How to do the task
 
Example:  
   
dice-3-md-12-ii.png dice-1-md-120.png dice-3-md-12-ii.png
    3         1        3
 i.e., a pair of 3's
 
In this task you will do an experiment to give an estimate of the probability of getting a pair of the same numbers when three dice are thrown. Do not count triples (three of the same number).
 
a)
Predict the probability of getting a pair of the same number ____________________
 
b)
You are to complete the table below.
The first three lots of 10 throws have been done for you and recorded in the table. In the first 10 throws there were 7 pairs, in the second 10 throws there were 6 pairs, and in the third 10 throws there were 4 pairs. 
 
You are to throw and record the next seven lots of 10 throws by following these instructions:
  1. Roll the three dice 10 times and count the number of times there is a pair. Record the number of pairs in the column labelled "Number of pairs in these 10 throws".
  2. Add this number to the previous number in the "Total number of pairs" column, and enter this in the "Total number of pairs" column.
  3. Divide the "Total number of pairs" by the "Total number of throws" and enter the result in the "Cumulative probability of a pair" column.
 
 
 
Complete the table below for the next seven lots of 10 throws of the dice (you do not have to write the numbers in brackets in the third column which are just there to help you).
 
Total number of throws
Number of pairs in these 10 throws  
(Instruction 1)
Total number of pairs
(Instruction 2)
Cumulative probability of a pair  
(Instruction 3)
10
7
7
7/10 = 0.7
 20
6
13 (7 + 6)
13/20 = 0.65
 30
4
17 (13 + 4)
17/30 = 0.57
 40
    
 
 50
    
 
 60
    
 
 70
    
 
 80
    
 
 90
    
 
 100
    
 
 

c)
Using the table in b), what is the probability of throwing a pair after a total of
 
  i)
20 throws? __________
 
  ii)
100 throws? __________
 
d) i)
Which would be the better number of throws to use to get the best estimate the probability of throwing a pair. (Circle one) 

(A) 20 throws
(B) 50 throws
(C) 100 throws
(D) 200 throws
(E) They are all just as good.
 
  ii)
Explain your answer.
 
 
 
 
 
 
 
 
Task administration: 
This task is completed with pencil and paper, and other equipment.
[Equipment: Three dice; calculator.]
  • This task is completed with pencil and paper, and other equipment.
  • Discuss the task and how each of the first three rows of the table have been filled out using the instructions in part a). These instructions could be put on an OHT.
  • When all students have finished the task, tell them the theoretical probability for getting a pair with three fair dice. (See scoring page.)
Level:
5
Curriculum info: 
Maths, Statistics, Probability
Key Competencies: 
Thinking, Using language, symbols, and texts
Keywords: 
long-run relative frequency, probability, prediction, experiments
Description of task: 
Students count the number of pairs thrown in a series of throws of three dice to evaluate the long-run relative frequency of getting a pair.
Curriculum Links: 
Key competencies
This resource involves predicting the outcome of a probability situation, and communicating the findings of the probability experiment, and explaining variation. These relate to the Key Competencies: Thinking, and Using language, symbols and text.

For more information see http://nzcurriculum.tki.org.nz/Key-competencies

Learning Progression Frameworks
This resource can provide evidence of learning associated with
Interpreting statistical and chance situations, set 5
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
1 mark
 
1 mark
 
2 marks (for all correct)
or
1 mark (for 1-2 errors)
 a)  

Numbers from 0-10 recorded in all spaces of "Number of pairs in these 10 throws" column. "Total number of pairs" column completed correctly.

"Cumulative probability of a pair" column completed correctly.
 
1-2 errors in the "Cumulative probability of a pair" column.
NOTE: Accept probabilities written as decimals or percentages.
1 mark
1 mark		 b) i)
ii)
13/20 [Accept written as decimal or percentage]
Correct answer consistent with the answers recorded in the last row of table.

1 mark
1 mark

 c) i)
ii)

D
The more throws, the more accurate the estimate or similar answer.

NOTE: The true probability for fair dice of throwing a pair with 3 dice is: 90 / 216 = 15/36 = 5/12 = 0.417 (to 3 d.p.)