Four dice game I

Four dice game I

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

Practical task
In a game four dice are thrown and the numbers facing up are added together.

Example:

  dice-2-md-120.png dice-3-md-120.png. dice-5-md-120.png  dice-4-md-120.png
Sum = 14 = 2 + 3 + 5 + 4

  • Player 1 wins if the sum is 11 to 17.
  • Player 2 wins if the sum is 4 to 10 or 18 to 24.
a) i)
Which player do you think is the most likely to win?    
 
Player 1       Player 2      They are equally likely       (circle one)
 
  ii)
Explain your answer to a) i)
__________________________________________________________________

__________________________________________________________________
 
b)
You are to carry out this experiment in pairs. Throw four dice and sum the numbers that are facing up. Record which player wins in the table below. Do this 100 times.
 
 
c) i) What is the probability of Player 1 winning based on your results for b) above? __________
  ii)
How could you get a more accurate estimate of the probability that Player 1 wins?
 
__________________________________________________________________
 
__________________________________________________________________
 
d) i) Do the results of your experiment indicate the game is fair?

Yes   /   No     (Circle one)
  ii)
Explain your answer to d) i).
 
__________________________________________________________________
 
__________________________________________________________________

 

Task administration: 

This task is completed with pencil and paper, and other equipment.

Equipment:

Four dice.

Students work in pairs for part b) only.

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, peer assessment
Description of task: 
Pairs of students play a game with four dice, calculate the probability of winning the game, comment on how to get a more accurate estimate of the probability, and explain whether the game is fair based on 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 tothe Key Competencies: Relating to others.

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)		 Accept any of the circled options with a reasonable explanation even if it is incorrect.
b)  

Systematic tally marks used e.g., rather than and frequencies given corresponding to tally marks.
Frequencies add to 100.
c)

i)
 
ii)

Probability correctly based on results for a).
NOTE: Accept if written as a fraction, decimal, orpercentage.
Increase the number of throws or equivalent statement.
d)

i)
ii)

 No
Any 1 of:

  • Player 1 won more often in the experiment.
  • There are more ways of getting the middle numbers (11-17).
  • Player 1's numbers are more likely.
  • A fair game has a 0×5 (approximately) chance of each player winning (and in this game it doesn't).
  • Equivalent statements.

NOTE: Accept, c) i) "Yes" as long as the number of times Player 1 wins is close to the number of times Player 2 wins (see notes below), and if a reasonable statement is made in c) ii), e.g., "Each player won about the same number of times." or "Player 1 has fewer numbers but they are more likely, i.e., the probabilities balance each other".

NOTE: The probability of Player 1 winning is 892/1296 = 0.688

  1. With 100 throws of 4 fair dice the estimate of the probability of Player 1 winning will be between 0×60 and 0×78 about 95% of the time. So in almost all cases, Player 1 will win far more often (i.e., wins 60% or more games 97×5% of the time) indicating the game is unfair.
  2. If a fair game is played 100 times between two players, the estimate of the probability of Player 1 winning is between 0×4 and 0×6 about 95% of the time. So only about 2×5% of the time will Player 1 seem to have a big advantage (i.e., wins 60% or more games).