Two dice game I

Two dice game I

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about predicting, then recording, the outcome of a game of chance.
Practical task

For Game A and Game B you are to:
  • decide if each game is fair or not and explain your answer;
  • play each game in pairs, but individually record and explain your results. 
Game A: Throw two dice and add the top two numbers together. 

  • Player 1 wins when the result is an odd number.
  • Player 2 wins when the result is an even number.
 dice-4-md-120.png dice-2-md.png
Sum = 6
a) i)
Using the table below, who is more likely to win this game? (Circle one)

(A) Player 1         (B) Player 2         (C) Both players are equally likely.
 
  ii)
Explain your answer.
 
 
 
 
 

Sum of two dice 2 3 4 5 6 7 8 9 10 11 12
Number of ways it can occur 1 2 3 4 5 6 5 4 3 2 1
 
Game B: Throw the two dice and multiply the top two numbers together.

  • Player 1 wins when the result is an odd number.
  • Player 2 wins when the result is an even number.
 ​ ​dice showing 4 ​ ​dice showing 2​
Product = 8
b)
i)
Who is more likely to win this game? (Circle one) 

  (A) Player 1         (B) Player 2         (C) Both players are equally likely.
 
  ii)
Explain your answer.
 
 
 
 
 
 
 
  • Pair up with someone else who has finished a) and b).
  • Get two dice and complete parts c) and d).
 
c)
 
i)
 
Play Game A 50 times and record your results in the table below.
 
   
Tally Frequency
Player 1 wins  
 
 
Player 2 wins  
 
 
 
 
ii)
 
Do your results suggest that Game A is fair?   Yes   /   No     (Circle one)
 
Explain your answer.
 
 
 
 
 
 
 
d)
 
i)
 
Play Game B 50 times and record your results in the table below.
 
   
Tally Frequency
Player 1 wins  
 
 
Player 2 wins  
 
 
 
 
ii)
 
Do your results suggest that Game B is fair?   Yes   /   No     (Circle one)
 
Explain your answer.  
 
 
 
 
 
 
Task administration: 
Equipment     Two dice.
 
Administration of task
  • Students complete parts a) and b) by themselves without dice.
  • Put students into pairs to play the games in c) and d), giving them two dice per pair.
  • Students should each fill in their own worksheet.
Levels:
4, 5
Curriculum info: 
Maths, Statistics, Probability
Key Competencies: 
Relating to others, Thinking, Using language, symbols, and texts
Keywords: 
probability, fair games, prediction, experiments, predict-observe-explain
Description of task: 
Students predict whether two dice games are fair and give their reasons. They then play each game (investigate) fifty times and explain their results.
Curriculum Links: 
This resource can be used to help to identify students' understanding of predicting the results of a chance situation, and checking consistency between experimental and theoretical results (including models of all possible outcomes).
 
Key competencies
This resource involves:
  • Predicting the outcome of a probability situation, which relates to the Key Competency: Thinking
  • Communicating the findings of the probability experiment, and explaining  variation, which relates to the Key Competency: Using language, symbols and text
  • Working collaboratively with another student, which relates to the Key Competencies: Relating to others

This resource involves predicting the outcome of a probability situation, and communicating the findings of the probability experiment, which 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, sets 4-5
Interpreting statistical and chance situations, set 5
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
a)

i)
ii)

C
There are the same number (18) of ways an even number can be thrown as there are ways an odd number can be thrown.
b) i)
ii)		 B
Accept any 1 of:

  • There are far more ways an even number can come up than the number of ways an odd number can come up (27 even compared with 9 odd).
  • You can only get an odd number if both dice are odd.
  • Equivalent statements.
c)

i)
ii)

 Game A played 50 times and the tally and frequency columns completed consistent with tally.

  • Yes
  • A reasonable statement consistent with their actual frequencies given, e.g., "The game is fair because each player won it about the same number of times." or "It is just luck that one player won (a few times) more often". NOTE: Accept if one player wins far more often than the other, and they answer "No" and a reasonable answer is given. (See Note 1 below.)
d)

i)
ii)

 Game B played 50 times and the tally and frequency columns completed.

  • No
  • A reasonable statement consistent with their actual frequencies given, e.g., "The game is unfair because Player 2 won it far more often". NOTE: Accept if each player wins about the same number of times, and they answer "Yes" and a reasonable answer is given. (See Note 2 below.)

 

NOTE:
  1. If the games are fair, one player will win between 16 to 34 times out of 50 games, 95% of the time. Either player winning fewer than 16 times indicates the game is not fair. Students are more likely to want the wins to fall into the range of 20 to 30 wins out of 50 to conclude the game is fair.
  2. In Game B, Player 2 should win 75% of the time. Player 2 will win between 30 and 44 times in 95% of sets of 50 games. Any result from 34 and upwards hence indicates an unfair game, with results from 30 to 33 wins for Player B being ambiguous. Accept the response "No" for results of 30 or more wins for B, and "Yes" for fewer than 30 wins for B.

EXTENSION: Pool the results for all groups and get students to revisit their responses to c) ii) and d) ii).

 

Diagnostic and formative information: 
 

Common error

Common reason

a) i)

Player 2

Because there are more even numbers than odd numbers (6 compared with 5).

b) ii)

18 even, 9 odd P(even) = 2/3

Counts e × e = e, o × o = o, o × e = e, but ignores e × o = e.