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Director  Joined: 07 Nov 2004
Posts: 502
What is the probability of getting a sum of 8 or 14 when  [#permalink]

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9
44 00:00

Difficulty:   95% (hard)

Question Stats: 32% (03:26) correct 68% (03:04) wrong based on 602 sessions

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What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

A. 1/6
B. 1/4
C. 1/2
D. 21/216
E. 32/216
Intern  Joined: 16 Nov 2004
Posts: 18
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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14
18
Quote:
"Gayathri"
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

a) 1/6
b) 1/4
c) 1/2
d) 21/216
e) 32/216

Is there a shorter way to do this than listing each possibility?

I am confident that there is a better way, but I looked at the first few possibilities

throw a 3 - 1 way
throw a 4 - 3 ways
throw a 5 - 6 ways
throw a 6 - 10 ways

and recognized these as triangular numbers (the series could also be identified as add 2, add 3, add 4 ...).

from there it was easy to calculate that there are 21 ways to hit an 8 and 15 ways to hit a 14. (Problems with fair dice and coins produce symmetrical probability distributions, so one can count down from 1 way to throw an 18)

[more than 2 minutes - less than 3 minutes]
##### General Discussion
Manager  Joined: 31 Aug 2004
Posts: 118
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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3
Richie Weasel wrote:
Quote:
"Gayathri"
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

a) 1/6
b) 1/4
c) 1/2
d) 21/216
e) 32/216

Is there a shorter way to do this than listing each possibility?

I am confident that there is a better way, but I looked at the first few possibilities

throw a 3 - 1 way
throw a 4 - 3 ways
throw a 5 - 6 ways
throw a 6 - 10 ways

and recognized these as triangular numbers (the series could also be identified as add 2, add 3, add 4 ...).

from there it was easy to calculate that there are 21 ways to hit an 8 and 15 ways to hit a 14. (Problems with fair dice and coins produce symmetrical probability distributions, so one can count down from 1 way to throw an 18)

[more than 2 minutes - less than 3 minutes]

Wow, Richie, this is a fast and easy way to solve it! I will take a note of this method! Manager  Joined: 30 Jul 2004
Posts: 67
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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Is there any other to solve this question ?

Can you elaborate the answer pls. ?
Director  Joined: 07 Nov 2004
Posts: 502
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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4
4
gmat+obsessed wrote:
Is there any other to solve this question ?

The other way would be to list out each option.
The options for a sum of 8: (6,1,1) has 3 options ie (6,1,1), (1,6,1), (1,1,6); (5,2,1) has 6 options, (4,3,1) has 6 options, (4,2,2) has 3 options, (3,3,2) has 3 options. We have 21 options to get 8.

The options for a sum of 14: (6,4,4) has 3 options, (6,5,3) has 6 options, (6,6,2) has 3 options, (5,5,4) has 3 options. We have 15 options to get 14.

Total: 21+15= 36/216 = 1/6.
Director  G
Joined: 23 Jan 2013
Posts: 519
Schools: Cambridge'16
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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2
I missed some options for 8 as a sum. Idea is that options containing three different number have 3!=6 possible events, options containing 2 identical and 1 other have 3!/2!*1!=3 possible events. By multiplying and summing we get the answer
Intern  Joined: 15 Nov 2012
Posts: 2
What is the probability of getting a sum of 8 or 14 when  [#permalink]

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5
3
Here's another way:
You start by listing out each possible triplet for every number on the dice for each of the two numbers we need (notice that for each triplet the probability is $$\frac{1}{6^3}$$):
8:
1:
[1,1,6]
[1,2,5]
[1,3,4]
[1,4,3]
[1,5,2]
[1,6,1]

2:
[2,1,5]
[2,2,4]
[2,3,3]
[2,4,2]
[2,5,1]

…and so on...

Soon thereafter, you realize that, for each number of the dice, there is one triplet less that adds up to 8, than the previous number (i.e. 1-triplets: 6, 2-triplets: 5, 3-triplets: 4, and so on…). Given that $$\frac{1}{6^3}$$ is a common factor to all triplets, we get that: $$\frac{1}{6^3}*(6+5+4+3+2+1)$$ or $$\frac{1}{6^3}*(21)$$.

Then we do a similar process for 14:

1:
(No possible combination adds up to 14)

2:
[2,6,6]

3:
[3,5,6]
[3,6,5]

…and so on…

So you'll notice that a similar thing happens in this case: from 2 on, for each number of the dice, there is one triplet more that adds up to 14, than the previous number (i.e. 2-triplet: 1, 3-triplets: 2, 4-triplets: 3, and so on…). Again, given that $$\frac{1}{6^3}$$ is a common factor to all triplets, we get that: $$\frac{1}{6^3}*(1+2+3+4+5)$$ or $$\frac{1}{6^3}*(15)$$.

Given that we need the probability of getting a sum of 8 OR 14, we add up both of these cases:
$$\frac{1}{6^3}*(21)+\frac{1}{6^3}*(15)$$

We factor out $$\frac{1}{6^3}$$, and find that:
$$\frac{1}{6^3}*(36)$$, and by recognizing that $$36 = 6^2$$, we cross it out with $$6^3$$ to find that the probability of getting a sum of 8 or 14 when rolling three fair dice is $$\frac{1}{6}$$.

Manager  S
Joined: 06 Oct 2015
Posts: 86
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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1
Richie Weasel wrote:
Quote:
"Gayathri"
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

a) 1/6
b) 1/4
c) 1/2
d) 21/216
e) 32/216

Is there a shorter way to do this than listing each possibility?

I am confident that there is a better way, but I looked at the first few possibilities

throw a 3 - 1 way
throw a 4 - 3 ways
throw a 5 - 6 ways
throw a 6 - 10 ways

and recognized these as triangular numbers (the series could also be identified as add 2, add 3, add 4 ...).

from there it was easy to calculate that there are 21 ways to hit an 8 and 15 ways to hit a 14. (Problems with fair dice and coins produce symmetrical probability distributions, so one can count down from 1 way to throw an 18)

[more than 2 minutes - less than 3 minutes]

Hi,
how do you throw 3 in 1 way? I have not understood these points. Will you explain these?
Intern  Status: Bring it on.....!
Joined: 11 Jul 2013
Posts: 17
Location: India
Concentration: Technology
GMAT 1: 660 Q49 V31
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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1
You can throw 3 dices simultaneously to get sum as 3 in only one way - i.e. each dice should have 1 . Hence , you throw 3 in 1 way. And similarly for other numbers.

NaeemHasan wrote:
Hi,
how do you throw 3 in 1 way? I have not understood these points. Will you explain these?

_________________
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Manager  S
Joined: 06 Oct 2015
Posts: 86
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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nick28 wrote:
You can throw 3 dices simultaneously to get sum as 3 in only one way - i.e. each dice should have 1 . Hence , you throw 3 in 1 way. And similarly for other numbers.

NaeemHasan wrote:
Hi,
how do you throw 3 in 1 way? I have not understood these points. Will you explain these?

Now, got that. Can you describe about the triangular system as mentioned in the first reply?
Intern  B
Joined: 01 Sep 2016
Posts: 4
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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1
1
Richie Weasel wrote:
Quote:
"Gayathri"
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

a) 1/6
b) 1/4
c) 1/2
d) 21/216
e) 32/216

Is there a shorter way to do this than listing each possibility?

I am confident that there is a better way, but I looked at the first few possibilities

throw a 3 - 1 way
throw a 4 - 3 ways
throw a 5 - 6 ways
throw a 6 - 10 ways

and recognized these as triangular numbers (the series could also be identified as add 2, add 3, add 4 ...).

from there it was easy to calculate that there are 21 ways to hit an 8 and 15 ways to hit a 14. (Problems with fair dice and coins produce symmetrical probability distributions, so one can count down from 1 way to throw an 18)

[more than 2 minutes - less than 3 minutes]

How 15 ways to hit a 14 as per this logic?
Intern  Joined: 12 Jun 2014
Posts: 1
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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where did the 216 come from?
Intern  Joined: 23 Sep 2012
Posts: 2
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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1
Richie Weasel wrote:
Quote:
"Gayathri"
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

a) 1/6
b) 1/4
c) 1/2
d) 21/216
e) 32/216

Is there a shorter way to do this than listing each possibility?

I am confident that there is a better way, but I looked at the first few possibilities

throw a 3 - 1 way
throw a 4 - 3 ways
throw a 5 - 6 ways
throw a 6 - 10 ways

and recognized these as triangular numbers (the series could also be identified as add 2, add 3, add 4 ...).

from there it was easy to calculate that there are 21 ways to hit an 8 and 15 ways to hit a 14. (Problems with fair dice and coins produce symmetrical probability distributions, so one can count down from 1 way to throw an 18)

[more than 2 minutes - less than 3 minutes]

When you say that these are triangular numbers, the series is as follows:

Total number - No. of ways
3 1
4 3
5 6
6 10
7 15
8 21
9 28
10 36
11 36
12 28
13 21
14 15
15 10
16 6
17 3
18 1

Adding them all, total no. of ways : 240

But for 3 throws of a dice, total no. of ways = 6x6x6 = 216

How do you account for the discrepancy?
Intern  B
Joined: 12 Jan 2017
Posts: 10
Location: United States
GMAT 1: 740 Q50 V40
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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gayathri wrote:
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

A. 1/6
B. 1/4
C. 1/2
D. 21/216
E. 32/216

If we're rolling 3 dice simultaneously, why is it that we're counting, for example, (6,1,1) as 3 ways?
Is it assumed that 3 dice are distinguishable?
Math Expert V
Joined: 02 Sep 2009
Posts: 60555
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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sharkr wrote:
gayathri wrote:
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

A. 1/6
B. 1/4
C. 1/2
D. 21/216
E. 32/216

If we're rolling 3 dice simultaneously, why is it that we're counting, for example, (6,1,1) as 3 ways?
Is it assumed that 3 dice are distinguishable?

Yes. Consider the die to be red, blue and green. Then 6, 1, 1 can occur in 3 ways:

red - blue - green
6 - 1 - 1
1 - 6 - 1
1 - 1 - 6.
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Director  S
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 520
Location: India
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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Possible outcomes (1,1,6): 3 ways, (1,2,5): 6 ways, (1,3,4): 6 ways, (2,2,4): 3 ways, (2,3,3): 3 ways, (4,4,6): 3 ways, (4,5,5): 3 ways, (5,6,3): 6 ways, (6,6,2): 3 ways
Total outcomes =36
Total outcomes = 6*6*6 = 216
Probability = 36/216 = ⅙ . Option A
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GMAT Mentors Intern  Joined: 23 Jul 2017
Posts: 1
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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I dont understand why there are only three ways to roll (1,1,6). There are three difference dices; therefore there is supposed to be 6 ways.
Math Expert V
Joined: 02 Sep 2009
Posts: 60555
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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Snooopy wrote:
I dont understand why there are only three ways to roll (1,1,6). There are three difference dices; therefore there is supposed to be 6 ways.

_________________
Intern  B
Joined: 28 Apr 2015
Posts: 6
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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Snooopy wrote:
I dont understand why there are only three ways to roll (1,1,6). There are three difference dices; therefore there is supposed to be 6 ways.

1, 1, 6
1, 6, 1
6, 1, 1

3 ways
_________________
Dan Morgan
MBA Wisdom
http://www.mbawisdom.com
Intern  B
Joined: 28 Apr 2015
Posts: 6
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

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1
gayathri wrote:
What is the probability of getting a sum of 8 or 14 when rolling 3 dice simultaneously?

A. 1/6
B. 1/4
C. 1/2
D. 21/216
E. 32/216

1 Dice: Probabilities of sums

1: 1/6
2: 1/6
3: 1/6
4: 1/6
5: 1/6
6: 1/6

2 Die: Probabilities of sums

2: 1/36
3: 2/36
4: 3/36
5: 4/36
6: 5/36
7: 6/36
8: 5/36
9: 4/36
10: 3/36
11: 2/36
12: 1/36

Probability of 8 with 3 die: 1 on first then 7 on next two, 2 on first and 6 on next two etc...
Probability of 8 with 3 die = (1/6)[(6 + 5 + 4 + 3 + 2 + 1)/36] = 21/216

Probability of 14 with 3 die: 2 on first then 12 on next two, 3 on first and 11 on next two etc...
Probability of 14 with 3 die = (1/6)[(5 + 4 + 3 + 2 + 1)/36] = 15/216

Probability of 8 OR 14 with 3 die = 21/216 + 15/216 = 36/216 = 1/6
_________________
Dan Morgan
MBA Wisdom
http://www.mbawisdom.com Re: What is the probability of getting a sum of 8 or 14 when   [#permalink] 09 Apr 2018, 11:15

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