An eight-sided dice

Probability Level 5

An eight-sided dice is rolled 3 times.
If the number obtained on the third roll is equal to the sum of the numbers obtained on the first roll and second roll, what is the probability that at least one of the rolls was a 2?

If the probability can be expressed as $\dfrac ab$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 10.

1 solution

Paul Fournier
Apr 15, 2016

Third roll: (first roll, second roll)

8 :(7,1) (1,7) (6,2) (2,6) (5,3) (3,5) (4,4)
7:(6,1) (1,6) (5,2) (2,5) (4,3) (3,4)
6: (5,1) (1,5) (4,2) (2,4) (3,3)
5: (4,1) (1,4) (3,2) (2,3)
4: (3,1) (1,3) (2,2)
3: (2,1) (1,2)
2 :(1,1)
Of the 28 possibilities 12 have at least one 2
12/28 = 3/7 and a+b=10

By your logic the roller of the die is twice as likely to roll a 7 and a 1 than they are to roll a 1 and a 1. Logically this is not true. If this were a problem that required only discrete outcomes to be considered then you'd have to rephrase the problem such that "if, on the second roll, the die shows the same number as the first die, reroll"

Mike Ramia - 5 years, 1 month ago

Correct solution: When rolling two die of 8 sides each, the number of possible outcomes (non-distinct) is 64. Which gives us the denominator. Meanwhile, the number of combinations with at least one 2 rolled is 24. 24/64 simplifies to 3/8. X=a+b X=11

Mike Ramia - 5 years, 1 month ago

Should we not apply Bayes theorem here... i.e partitioning the set of sum of 2 rolls equals thord as having the cause of a single 2 and a double 2 and triple 2 and no 2 at all?

Milind Blaze - 5 years, 1 month ago

I see only 11, where is the 12th?

James Simpson - 5 years, 2 months ago

James,

You may have forgotten to count when the third roll is a 2 and and the first and second are 1s.

Paul Fournier - 5 years, 2 months ago

This answer is wrong. One has to count (4,4) two times for the 3rd roll being 8 and so for all other turns whose numbers for the first two rolls are the same.

Bernardo Tavora - 5 years, 2 months ago

Bernardo,
Rather then calling them the first roll and second roll one could use a red dice and a blue dice.
(red 5, blue 3) and (red 3, blue 5 ) are different but (red 4, blue 4) and (red 4, blue 4) are the same.

Paul Fournier - 5 years, 2 months ago

An eight-sided dice

The answer is 10.

1 solution

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