Dice Challenge
For this investigation you need 3 dice and a partner.
Dice A is labelled 2 2 2 2 6 6
Dice B is labelled 1 1 5 5 5 5
Dice C is labelled 3 3 3 4 4 4
With your partner, play a game with the following rules:
1.The first player chooses a dice.
2.The second player chooses a die from the remaining two dice.
3.Each player rolls his or her die and the player with the higher number wins a point.
4.Repeat this with the same dice 12 times. The player with the most points wins the game.
With these rules is it possible for one player to win nearly all the time?
If this is so how is it possible?
After 20 minutes or so each pair came up with the result that in order to win you must be the second player, ie: the one that chooses from two, not three, dice.
But this made no sense to me.
If Dice A beats Dice B and Dice B beats Dice C, how is it possible for Dice C to beat Dice A? But it does!! And I can prove it!!!
All this reminded me of a similar problem that I had had in my notebook for a number of years and to which I had no satisfactory solution.
The problem was to select a dice from a set of four with the best possible chance of winning over any of the others. Of course the dice are marked up in odd ways:
Dice A: 3 3 3 3 3 3
Dice B: 2 2 2 2 6 6
Dice C: 1 1 1 5 5 5
Dice D: 4 4 4 4 6 6
Champion Dice
An ordinary dice has 21 spots in total, 1+2+3+4+5+6=21. I want you to design an alternative dice with a total of 21 spots that will beat mine in a battle. So for example you could label it 2+2+3+4+5+5=21. Test your ideasout and give clear reasons for you final decision.
Introductory Activity
Equipment:30 plastic cups with lids, each lid should have a hole in it, large number of black, white and red balls.
Prepare 30 plastic cups with a mixture of black, white and red balls in each 10 in total. Tell the students that their task is to identify how many of each colour there are in their cup. They can withdraw one marble at a time look at it then they have to replace it. They can do this as often they like.
Before bringing the group together ask them to check their prediction.
Main Activity 1.
Equipment:70 dice, little labels to stick on the dice to re-number them.
An ordinary dice has 21 spots in total, 1+2+3+4+5+6=21. I want you to design an alternative dice with a total of 21 spots that will beat mine in a battle. So for example you could label it 2+2+3+4+5+5=21. Work in pairs or threes to share ideas and test those ideas out.
Group Session:
Ask someone for his or her suggestion for a challenger - test it in front of the group for 10 times, 20 times, 30 times. Discuss outcomes, how many trials are needed, do we agree that this is better?
How can we test this theoretically rather than practically?
List outcomes:
Table of outcomes:
Show that the theoretically probability of the challenger winning is 0.5 and of the champion retaining their crown is 0.5. Ask for another suggestion and see whether this has a better chance of winning.
Main Activity 1+
Send them back to work on finding the dice with the best chance of challenging the champion. Bring them back to if anyone has a challenger. - Conclude that all dice with 21 spots have an equal chance of winning.
Main Activity 2
Equipment:4 coins per pair with stickers on so that they can be numbered.
Adapt the previous activity so that each pair has 2 coins with the 4 sides numbered with a total of 10 dots. You multiply your 2 numbers to get your score.
So person A numbers the coins 1,3 and 4,2 and person B 2,5 and 2,1
So A can score 4,6,2,12 and B can score 4,2,10,5 so P(A winning) = 6/16 and
P(B winning) = 8/16
A/B / 4 / 2 / 10 / 52 / B / - / B / B
4 / - / A / B / B
6 / A / A / B / A
12 / A / A / A / A
Main Activity 3
Equipment:12 cups, one with 50 balls in, the others with 20 in each.
20 copies of the table to fill in results.
Discuss when it might be necessary to take samples similar to the introductory exercise: Whenever surveying public opinion. Give some examples of surveys and briefly discuss problems with the surveying technique. [See OHP]
Place in a bag a mixture of black, red and white balls to symbolise the ratio of voters for the labour, conservative and liberal democrats.
(50 balls in total)
Ask one voter what he/she voted. What can we conclude from this?
Ask 9 more voters what can we now conclude?
Ask 10 more voters etc…
Mark in the table below our estimate for the percentage voting for each of the parties.
Labour / Conservative / Lib DemNumber of people surveyed / Red / White / Black
Voters / % / Voters / % / Voters / %
10 / 111 / 30 / 1111 / 40 / 111 / 30
3 / 4 / 3
20 / 11111 / 40 / 1 / 25 / 1111 / 35
8 / 5 / 7
30 / 1111 / 40 / 11111 / 33 / 1 / 27
12 / 10 / 8
40
What do we expect to happen as we increase the number of voters we ask?
Have 12 bags, to represent 12 counties, prepared with 20 balls in each. Seat the students in groups again and ask them to complete the table below and make their own prediction for what percentage voted for each of the parties.
Put your results into this table.
Labour / Lib Dem / ConservativeNumber of people surveyed / Red / Black / White
Voters / % / Voters / % / Voters / %
10
20
30
40
50
60
70
80
90
100