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

Question

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

mst · Answer

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.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards