Higher Chance

Probability Level 1

If I have 3 dice, which of the following probabilities is not equal to the probability of rolling a pair of sixes with two dice?

A) With 3 dice, rolling any triple (i.e. 1, 1, 1)

B) With 3 dice, rolling to get a sum of 16

C) With 3 dice, rolling two equal numbers and a differing one (i.e. 1, 2, 1)

D) With 3 dice, rolling to get a sum of 5

D B A C

Omri Weininger by Omri Weininger , 16, Australia

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1 solution

Omri Weininger
Apr 27, 2019

The probability of rolling a pair of sixes with two dice is 1 36 \frac{1}{36}

Including permutations, there are 216 different outcomes of rolling 3 dice and we'll use this to compare the outcomes:

There are only 6 possible triples that can be rolled with three dice and 6 216 \frac{6}{216} is equal to 1 36 \frac{1}{36}

With 16 there are again only 6 ways to roll:

  • 4, 6, 6

  • 5, 5, 6

  • 5, 6, 5

  • 6, 4, 6

  • 6, 5, 5

  • 6, 6, 4

And with 5 there are also only 6 ways to roll:

  • 1, 1, 3

  • 1, 2, 2

  • 1, 3, 1

  • 2, 1, 2

  • 2, 2, 1

  • 3, 1, 1

When it comes to C, though, there are 60 different possibilities which I can't list here but is provable: Once you have rolled your first number i.e. 5, there are two ways to get this type of roll, either a) the next two numbers are a double, excluding 5, which gives 5 possibilities, or b) the next two numbers are different and one of them is a 5, which gives another 5 possibilities. Together these make 10, and if you multiply these results by 6 then you have 60.

0 Helpful 1 Interesting 0 Brilliant 0 Confused

I agree with the answer, but I dont agree with the 60 for B (Which I think you meant C but I digest).
Its either 30 if order matters or 90 if order doesnt matter.
I think where you went wrong was "the next two numbers are different and one of them is a 5, which gives another 5 possibilities." as this should give you an additional 10 possibilities (either of the two dice can be the 5)

Heres the 30 ordered:
1 1 2
1 1 3
1 1 4
1 1 5
1 1 6
1 2 2
1 3 3
1 4 4
1 5 5
1 6 6
2 2 3
2 2 4
2 2 5
2 2 6
2 3 3
2 4 4
2 5 5
2 6 6
3 3 4
3 3 5
3 3 6
3 4 4
3 5 5
3 6 6
4 4 5
4 4 6
4 5 5
4 6 6
5 5 6
5 6 6

Heres the 90 unordered:
1 1 2
1 1 3
1 1 4
1 1 5
1 1 6
1 2 1
1 2 2
1 3 1
1 3 3
1 4 1
1 4 4
1 5 1
1 5 5
1 6 1
1 6 6
2 1 1
2 1 2
2 2 1
2 2 3
2 2 4
2 2 5
2 2 6
2 3 2
2 3 3
2 4 2
2 4 4
2 5 2
2 5 5
2 6 2
2 6 6
3 1 1
3 1 3
3 2 2
3 2 3
3 3 1
3 3 2
3 3 4
3 3 5
3 3 6
3 4 3
3 4 4
3 5 3
3 5 5
3 6 3
3 6 6
4 1 1
4 1 4
4 2 2
4 2 4
4 3 3
4 3 4
4 4 1
4 4 2
4 4 3
4 4 5
4 4 6
4 5 4
4 5 5
4 6 4
4 6 6
5 1 1
5 1 5
5 2 2
5 2 5
5 3 3
5 3 5
5 4 4
5 4 5
5 5 1
5 5 2
5 5 3
5 5 4
5 5 6
5 6 5
5 6 6
6 1 1
6 1 6
6 2 2
6 2 6
6 3 3
6 3 6
6 4 4
6 4 6
6 5 5
6 5 6
6 6 1
6 6 2
6 6 3
6 6 4
6 6 5

Kyle T - 2 years, 1 month ago

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