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A fair sided die labeled 1 to 6 is tossed three times. What

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A fair sided die labeled 1 to 6 is tossed three times. What [#permalink] New post 09 Feb 2012, 08:02
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A fair sided die labeled 1 to 6 is tossed three times. What is the probability the sum of the 3 throws is 16?
A) 1/6
B) 7/216
C) 1/36
D) 9/216
E) 11/216

I have a problem with these type if questions, especially if the sum is sth like 12,etc. (the number of combinations are large)

What is the way to solve this problem?

could somebody solve this using the expected value concept please?
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Re: Probability Dice problem [#permalink] New post 09 Feb 2012, 08:27
bsaikrishna wrote:
A fair sided die labeled 1 to 6 is tossed three times. What is the probability the sum of the 3 throws is 16?
A) 1/6
B) 7/216
C) 1/36
D) 9/216
E) 11/216

I have a problem with these type if questions, especially if the sum is sth like 12,etc. (the number of combinations are large)

What is the way to solve this problem?

could somebody solve this using the expected value concept please?


I would say first remember that the probability of anything is the number of outcomes that give you what you want divided by the total number of possible outcomes.

Now, the total number of possible outcomes is, of course, 6^3.

Fortunately, there are only two combinations of three dice throws that will give you a sum of 16:

5, 5, 6
4, 6, 6

(I have a hard time remembering a real GMAT question that would give you 12, as the number of potential combinations is indeed inordinately large and difficult to map out.)

However, our denominator counts out the number of throwing sequences (permutations) and not just combinations, so we will need to do the same. Effectively there are three throwing sequences for either combination:

6, 5, 5
5, 6, 5
5, 5, 6

4, 6, 6
6, 4, 6
6, 6, 4

So 6 throwing sequences we want divided by 6^3 throwing sequences we could get gives us 1/(6^2) or 1/36.

As for the "expected value," that isn't, unfortunately, of much help on this particular problem since all of the denominators in the answer choices are powers of 6.
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Re: Probability Dice problem [#permalink] New post 09 Feb 2012, 08:28
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bsaikrishna wrote:
A fair sided die labeled 1 to 6 is tossed three times. What is the probability the sum of the 3 throws is 16?
A) 1/6
B) 7/216
C) 1/36
D) 9/216
E) 11/216

I have a problem with these type if questions, especially if the sum is sth like 12,etc. (the number of combinations are large)

What is the way to solve this problem?

could somebody solve this using the expected value concept please?


Not sure what “expected value concept” is but used the following logic:

1. There are 2 ways to get 16 out of 3 throws: 6+6+4 and 6+5+5
2. There are 3 different ways to throw each of those combinations (6+6+4 or 6+4+6 or 4+6+6; the same is for 6-5-5), so we have 6 combinations to get 16
3. The probability of getting any number at a throw is 1/6, so the probability of throwing each combination is 1/6*1/6*1/6 = 1/216
4. 6*(1/216) = 1/36
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Re: Probability Dice problem [#permalink] New post 09 Feb 2012, 09:11
Expert's post
bsaikrishna wrote:
A fair sided die labeled 1 to 6 is tossed three times. What is the probability the sum of the 3 throws is 16?
A) 1/6
B) 7/216
C) 1/36
D) 9/216
E) 11/216

I have a problem with these type if questions, especially if the sum is sth like 12,etc. (the number of combinations are large)

What is the way to solve this problem?

could somebody solve this using the expected value concept please?


The sum of 16 is obtained if the dices show: (6, 5, 5) or (6, 6, 4).

Each can occur in 3!/2!=3 ways, for example we can get (6, 5, 5) in 3 ways: (6, 5, 5), (5, 6, 5), (5, 5, 6), permutation of three items (6, 6, 5) out of which 2 5's are identical.

So, # of favorable outcomes is 2*3=6;
Total # of outcomes is 6*6*6=216;

P=favorable/total=6/216=1/36.

Answer: C.

As for your question: generally if on the real test you are faced with some hard calculations, something like 1.79*4.89, then it might indicate two cases: either there is a shortcut solution for this problem, which won't involve such tedious math or answer choices are handy to deal with it.

Now, 12 can be obtained by combination of the following 6 cases: 1-5-6, 2-4-6, 2-5-5, 3-3-6, 3-4-5, 4-4-4 --> 3!+3!+3!/2!+3!/2!+3!+1=25 --> P=25/216. Since, it involves counting of 6 different cases and there is no real shortcut, then I doubt that you'll be asked about the probability of getting the sum of 12 (or the answers choices will be useful to find this value).

Having said that you should take into account the fact that the probability distribution in this question (and in many similar questions) is symmetrical and sometimes that could give a good shortcut. For example:
The probability of getting the sum of 3 (min possible sum) = the probability of getting the sum of 18 (max possible sum);
The probability of getting the sum of 4 = the probability of getting the sum of 17;
The probability of getting the sum of 5 = the probability of getting the sum of 16;
...
The probability of getting the sum of 10 = the probability of getting the sum of 11;

So, for our case, the probability of getting the sum of 16 is the same as the probability of getting the sum of 5 and you can find the combinations for the sum 5 instead: (2, 2, 1) or (3, 1, 1).

To summarize: combinations/probability questions can be solve in many ways, most of them are fairly straightforward, some have tricky shortcut and some can be solved wit the help of answer choices, so no hard math will be required.

Check similar question which involves expected value approach to practice: mary-and-joe-126407.html

Hope it helps.
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Re: A fair sided die labeled 1 to 6 is tossed three times. What [#permalink] New post 21 Jul 2013, 08:38
I would suggest better mug up following list for three dice.

3 4 5 6 7 8 9 10 | 11 12 13 14 15 16 17 18 <<< total
1 3 6 10 15 21 25 27 mirror 27 25 21 15 10 6 3 1 <<< possible cases.

1=1
1+2 = 3
3+3 = 6
6+4 = 10
10+5 = 15
15+6 = 21
following two not in pattern.
21+4 = 25
25+2 = 27
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Re: A fair sided die labeled 1 to 6 is tossed three times. What   [#permalink] 21 Jul 2013, 08:38
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