GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 31 Mar 2020, 21:07

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# What is the probability of getting a sum of 8 or 14 when

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 07 Nov 2004
Posts: 497
What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

26 Nov 2004, 06:31
9
44
00:00

Difficulty:

95% (hard)

Question Stats:

32% (03:26) correct 68% (03:04) wrong based on 604 sessions

### HideShow timer Statistics

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]

### Show Tags

26 Nov 2004, 06:51
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: 113
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

26 Nov 2004, 22:03
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]

### Show Tags

27 Nov 2004, 22:06
Is there any other to solve this question ?

Can you elaborate the answer pls. ?
Senior Manager
Joined: 07 Nov 2004
Posts: 497
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

28 Nov 2004, 05:05
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
Joined: 23 Jan 2013
Posts: 512
Schools: Cambridge'16
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

22 Oct 2014, 04:17
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]

### Show Tags

17 Nov 2014, 13:03
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
Joined: 06 Oct 2015
Posts: 86
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

26 Aug 2016, 10:10
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]

### Show Tags

27 Aug 2016, 20:54
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?

_________________
Live as if your were to die tomorrow. Learn as if you were to live forever - Gandhi
Manager
Joined: 06 Oct 2015
Posts: 86
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

30 Aug 2016, 07:08
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
Joined: 01 Sep 2016
Posts: 4
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

14 Sep 2016, 22:46
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]

### Show Tags

18 Sep 2016, 15:53
where did the 216 come from?
Intern
Joined: 22 Sep 2012
Posts: 2
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

30 Sep 2016, 13:48
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
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]

### Show Tags

03 Feb 2017, 04:50
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
Joined: 02 Sep 2009
Posts: 62380
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

03 Feb 2017, 06:10
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.
_________________
Director
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 515
Location: India
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

01 Mar 2017, 01:00
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
_________________
GMAT Mentors
Intern
Joined: 23 Jul 2017
Posts: 1
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

09 Aug 2017, 04:18
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
Joined: 02 Sep 2009
Posts: 62380
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

09 Aug 2017, 04:20
1
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
Joined: 28 Apr 2015
Posts: 6
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

09 Apr 2018, 09:59
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
Joined: 28 Apr 2015
Posts: 6
Re: What is the probability of getting a sum of 8 or 14 when  [#permalink]

### Show Tags

09 Apr 2018, 10:15
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, 10:15

Go to page    1   2    Next  [ 28 posts ]

Display posts from previous: Sort by