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# joe and moe throw 3 dice each. toe is the sum of points on

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Senior Manager
Joined: 11 May 2006
Posts: 258
joe and moe throw 3 dice each. toe is the sum of points on [#permalink]

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01 Oct 2006, 11:48
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joe and moe throw 3 dice each. toe is the sum of points on all 3 dice. If toe of moe is 10, what is the probability that joe will have more toe than moe ?

a) 24/64

b) 32/64

c)36/64

d)40/64

e)42/64
Manager
Joined: 01 Oct 2006
Posts: 242

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02 Oct 2006, 07:32
iced_tea wrote:

The possible â€˜toeâ€™ (sum) values for the 3 dices are 3 to 18.
Now, there is 1 way of getting toe = 3 ( 1,1,1)
3 ways of getting toe = 4 (1,1,2 : arranged in 3 ways for 3 dices)
6 ways of getting toe = 5 (1,2,2; 1,1,3 : arranged in 3 ways each)
:
3 ways of getting toe = 17 (6,6,5 : arranged in 3 ways)
1 way of getting toe = 18 (6,6,6)
If you notice the number of ways of getting toes is increasing in geometric progression. Similarly if you go in descending order from 18 , the number of ways again increase in geometric progression. And it seems the maximum ways of getting any sum is 18.

So the possible ways of getting toes = 3 to 18 are as follows:

Toe = 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Ways = 1 3 6 9 12 15 18 18 18 18 15 12 9 6 3 1

In order to win. Joe needs to get the sum on 3 dices between 11 to 18 (both inclusive). And as you can see there are 50% ways of doing that.

But I think there should be a better way of doing it, because it took me 20 minutes to just figure out the pattern for the number of ways of getting all toe values.
Re: PS - probability   [#permalink] 02 Oct 2006, 07:32
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