Five dice game I

Five dice game I

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Further Resources
This task is about predicting and calculating combinations.
Practical Task
 
dice-2-md-120.png
  
Example 2 3 1 3 4 One pair (of 3's)
 
When five dice are thrown these are the seven different types of number combinations they can have on them:
 
Combination Example
1. All dice different 1, 2, 4, 5, 6
2. One pair 1, 2, 3, 3, 4
3. Two pairs 2, 2, 5, 5, 6
4. One triple 1, 4, 4, 4, 5
5. One triple + one pair 2, 2, 3, 3, 3
6. Four dice the same 3, 6, 6, 6, 6
7. Five dice the same 4, 4, 4, 4, 4
 
a)
Using the combinations listed in the table, rank how often you would predict each one to occur, from most likely to least likely.
 
 
i)
most likely combination
____________________
 
 
ii)
 
iii)
 
iv)
 
v)
 
vi)
 

 
____________________
 
____________________
 
____________________
 
____________________
 
____________________
 
 
vii) 
 least likely combination 
____________________
 
 
 
b)
Throw the five dice 100 times and record your results in the table below.
You may work with a partner when you throw the dice.
 
 
Combination Tally Frequency
1.   All dice different    
2.   One pair    
3.   Two pairs    
4.   One triple    
5.   One triple + one pair    
6.   Four dice the same    
7.   Five dice the same    
 
c)
 
Use your results to calculate the probability of
 
 
i)
 
ii)
 
all dice different (i.e., combination number 1) __________
 
one pair (i.e., combination number 2) __________
 
d)
What would you do to get a more accurate estimate of all the probabilities?
 
 
 
 
 
Complete c) and d) by yourself and get your teacher to mark them.
 
e)
 
In the table below:
  • Record the results from at least four sets of 100 trials.
  • Add these up for each combination and put the total in the "Total'" column
 
Combination Group 1 Group 2 Group 3 Group 4 Total
1.   All dice different          
2.   One pair          
3.   Two pairs          
4.   One triple          
5.   One triple + one pair          
6.   Four dice the same          
7.   Five dice the same          
 
f)
Looking at the "Total" column, rank all the 7 combinations from the most likely to the least likely according to how often each type of combination actually occurred.
 
  i)
most common combination
 
 ____________________
 
ii)
 
iii)
 
iv)
 
v)
 
vi)
 
 
____________________
 
____________________
 
____________________
 
____________________
 
____________________
 
  vii)
least common combination 
 
 
____________________
 
g)
 
Compare the actual results in f) with what you predicted in a). Give the number(s) of the combination(s) for which your prediction was
 
 
i)
 
ii)
least accurate? ______________________________

most accurate? ______________________________
Task administration: 
This task is completed with pencil and paper, and other equipment.
[Equipment: Five dice.]
  • Hand out the sheets up to and including question d).
  • Work through a) with the whole group getting them to make quick intuitive predictions. Emphasise that there are no marks for these predictions.
  • Students can work in pairs for part b), but should do c) and d) independently.
  • Students work should be checked after part d).
  • The data of when five dice are thrown could be put onto display (or into a spreadsheet).
  • A class or group discussion should be held after part d). The way in which the results are accumulated may vary from four sets of 100 trials up to the whole classes' results.

Group/class discussion

  • Discuss with the class/group the answer to d), i.e., that the bigger the sample is, the better the estimate.
  • Hand out this second sheet with e) to g) on it.
  • Using the table below, accumulate the data for at least four sets of 100 trials, or for the whole class. Part e) may be done together on the white board, OHP, etc., with students copying down the results in the "Total" column.
Level:
5
Curriculum info: 
Maths, Statistics, Probability
Key Competencies: 
Relating to others, Thinking, Using language, symbols, and texts
Keywords: 
probability, long-run relative frequency, prediction, distribution, experiments
Description of task: 
Students predict the outcome of throwing five dice and confirm their predictions by conducting an experiment and calculating probabilities. After discussing how to get better estimates, they pool their results with others to get a more accurate estimate of how good their predictions were.
Curriculum Links: 
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 other students, 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, set 5
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 

 

  
No mark a)    

1 mark
1 mark

 b)  

Tallying done with all frequencies summed correctly.
Sum of the frequencies is 100.

2 marks
or
1 mark

 c)  

Both probabilities correctly calculated based on their results in b).
NOTE: Accept if written as a fraction, decimal, or percentage.
One probability correctly calculated.
1 mark d)   Increase the sample size or equivalent statements.
No mark e)    
1 mark f)  

All events ranked consistently with results in e). This will most likely be the following order: 

  1. One pair
  2. Two pairs
  3. One triple
  4. One triple + 1 pair
  5. All dice different
  6. Four dice the same
  7. Five dice the same

1 mark
 
 


1 mark

 g)

i)
 
 


ii)

Answer consistent with a) and f).
NOTE: Accept if at least one answer is correct and no incorrect answers are listed. Answer consistent with a) and f).
NOTE: Accept if at least one answer is correct and no incorrect answers are listed.

Give both marks if they were correct with all predictions and this is consistent with their response.

 

NOTE: The probabilities are:

All dice different

  120/1296

One triple + 1 pair

 150/1296
 

One pair

 600/1296

Four dice the same

 25/1296
 

Two pairs

 300/1296

Five dice the same

 1/1296
 

One triple

 200/1296    
 Many students will predict "All dice different" to be more likely than it actually is.