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Magoosh Brain Twister: A Trick Up His Sleeve – Explanation


Want to refresh your memory before we get started? Here’s Monday’s Brain Twister:


A conjuror will roll one red, six-sided die in his right hand and two blue, six-sided dice. What is the probability that the number on the red die will be greater than the sum of the two blue dice?

(A) 5/54
(B) 5/108
(C) 11/216
(D) 7/36
(E) 5/18

Answer and Explanation

A good place to start is to chart out how many different ways the red die can come up greater than the sum of the blue dice.

If red ‘1’ , zero possible ways red is greater than the blue
If red ‘2’, zero possible ways
If red ‘3’, one possible way (if you roll “snake eyes” with the blue dice)
If red ‘4’, three possible ways (1,1, 1,2 or 2,1)

At this point, you can see that all the previous cases will work when the red die is one higher.
So if red is ‘5’, then there are three new ways (1,3, 2,2, 3,1), which we add to the three ways identified when red is ‘4’. This gives us a total of ‘6’.

If red is ‘6’, we have four new ways for a total of 10.

This gives us:

1 + 3 + 6 + 10 = 20

Since, there are a total of 2^6 possibilities when rolling three die (6^3 = 216), the answer is 20/216 or 5/54, Answer: (A).


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