GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 13 Dec 2018, 00:41

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

Events & Promotions

Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• The winning strategy for 700+ on the GMAT

December 13, 2018

December 13, 2018

08:00 AM PST

09:00 AM PST

What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.
• GMATbuster's Weekly GMAT Quant Quiz, Tomorrow, Saturday at 9 AM PST

December 14, 2018

December 14, 2018

09:00 AM PST

10:00 AM PST

10 Questions will be posted on the forum and we will post a reply in this Topic with a link to each question. There are prizes for the winners.

You have two dice, what is the probability of rolling a sum of 3 BEFOR

Author Message
TAGS:

Hide Tags

Intern
Joined: 15 Apr 2010
Posts: 45
You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

06 Oct 2010, 00:21
1
1
00:00

Difficulty:

(N/A)

Question Stats:

50% (00:00) correct 50% (00:02) wrong based on 2 sessions

HideShow timer Statistics

You have two dice, what is the probability of rolling a sum of 3 BEFORE rolling a sum of 7?
Retired Moderator
Joined: 02 Sep 2010
Posts: 765
Location: London
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

06 Oct 2010, 00:47
catennacio wrote:

You have two dice, what is the probability of rolling a sum of 3 BEFORE rolling a sum of 7?

Thanks..

Can you define the problem more precisely ?

You start rolling the dice, and the sum of two dice is 3, but the third dice adds up to a 7 ?

Or do u roll three dice twice, and the first time you get a 3 and the second time a 7 ?
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 51160
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

06 Oct 2010, 02:39
1
catennacio wrote:

You have two dice, what is the probability of rolling a sum of 3 BEFORE rolling a sum of 7?

Thanks..

First of all please note that this question is far beyond the GMAT scope.

$$P(sum=3)=\frac{2}{36}=\frac{1}{18}$$: either (1,2) or (2,1) out of total of 36 different combinations of two dice;

$$P(sum=7)=\frac{6}{36}=\frac{1}{6}$$: (1,6), (2,5), (3,4), (4,3), (5,2), or (6,1);

$$P(other \ sum)=1-(\frac{1}{18}+\frac{1}{6})=\frac{7}{9}$$, probability of sums: 4, 5, 6, 8, 9, 10, 11, and 12;

Winning scenarios:
{sum=3} - we have sum of 3 on the first roll of two dice - $$P_1=\frac{1}{18}$$;
{other sum; sum=3} - on the first roll we have other sum and sum of 3 on the second roll - $$P_2=\frac{7}{9}*\frac{1}{18}$$;
{other sum; other sum; sum=3} - $$P_3=(\frac{7}{9})^2*\frac{1}{18}$$;
{other sum; other sum; other sum; sum=3} - $$P_4=(\frac{7}{9})^3*\frac{1}{18}$$;
...

So probability of rolling a sum of 3 BEFORE rolling a sum of 7 would be the sum of the above infinite series:
$$P=\frac{1}{18}+\frac{7}{9}*\frac{1}{18}+(\frac{7}{9})^2*\frac{1}{18}+(\frac{7}{9})^3*\frac{1}{18}+...=\frac{1}{18}(1+\frac{7}{9}+(\frac{7}{9})^2+(\frac{7}{9})^3+...)=\frac{1}{18}(1+\frac{\frac{7}{9}}{1-\frac{7}{9}})=\frac{1}{18}(1+\frac{7}{2})=\frac{1}{4}$$ (for geometric progression with common ratio $$|q|<1$$, the sum of the progression: $$b_1, b_2, ...$$ is $$Sum=\frac{b_1}{1-q}$$.).

OR:

As $$P(sum=3)=\frac{2}{36}$$ and $$P(sum=7)=\frac{6}{36}$$ then getting the sum of 7 is 3 times more likely than getting the sum of 3, so the sum of 3 has 1 chance out of 4 to get first out of any number of tries, so $$P=\frac{1}{4}$$ or $$P=\frac{\frac{2}{36}}{\frac{2}{36}+\frac{6}{36}}=\frac{1}{4}$$.

Hope it's clear.
_________________
Manager
Joined: 25 Jun 2010
Posts: 87
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

06 Oct 2010, 04:29
Excellently done Bunuel!

Posted from my mobile device
Intern
Joined: 15 Apr 2010
Posts: 45
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

06 Oct 2010, 15:42
Thank you very much Buneul for your answer. However I have a few questions:

I understand up to P(other sum). How can you conclude the winning scenario is other sum then sum 3? I know it's a possible case, but why is it (other sum, sum 3) will lead to a (other sum, sum 3, then sum 7). It could be (other sum, sum 3, sum 8) - after 3 rollings, while what we expect is sum 7. Therefore the (other sum, other sum,..., sum 3) could be a wrong case.

On the other hand, if we can conclude other sum, other sum,..., sum 3 is a winning scenario, have we assume the probability of sum 7 and other sum but not 7 are the same, while there are different?

I don't understand this sentence too "So probability of rolling a sum of 3 BEFORE rolling a sum of 7 would be the sum of the above infinite series" - how can you conclude it's the sum that series, how about if we have an AFTER word? What is the difference between BEFORE and AFTER?

Edit: I like your second approach, by comparing the fraction of the probability of 2 events we can say P = 1/4, but I still don't understand the word BEFORE.. What if they ask for AFTER? Will we still compare the 2 probabilities?
Math Expert
Joined: 02 Sep 2009
Posts: 51160
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

07 Oct 2010, 02:12
catennacio wrote:
Thank you very much Buneul for your answer. However I have a few questions:

I understand up to P(other sum). How can you conclude the winning scenario is other sum then sum 3? I know it's a possible case, but why is it (other sum, sum 3) will lead to a (other sum, sum 3, then sum 7). It could be (other sum, sum 3, sum 8) - after 3 rollings, while what we expect is sum 7. Therefore the (other sum, other sum,..., sum 3) could be a wrong case.

On the other hand, if we can conclude other sum, other sum,..., sum 3 is a winning scenario, have we assume the probability of sum 7 and other sum but not 7 are the same, while there are different?

I don't understand this sentence too "So probability of rolling a sum of 3 BEFORE rolling a sum of 7 would be the sum of the above infinite series" - how can you conclude it's the sum that series, how about if we have an AFTER word? What is the difference between BEFORE and AFTER?

Edit: I like your second approach, by comparing the fraction of the probability of 2 events we can say P = 1/4, but I still don't understand the word BEFORE.. What if they ask for AFTER? Will we still compare the 2 probabilities?

When we roll 2 dice the sum can be 3, 7, or some other number (not 3 and not 7). The questions asks for the cases when we get the sum of 3 BEFORE the sum of 7.

For example why the cases I wrote are the winning scenarios:
{sum=3} - means that right on the first throw we have sum of 3, so we have 3 before 7 (as no 7 at all);
{other sum; sum=3} - first roll not 7 and not 3, so we can continue. On the second throw we have sum of 3, so again 3 before 7 - OK;
{other sum; other sum; sum=3} - sum of 3 on the third roll;
{other sum; other sum; other sum; sum=3} - sum of 3 on the fourth roll;
....
{other sum; other sum; other sum; ..., other sum on the nth roll; sum of 3 on the (n+1)th roll} - out of n+1 rolls we have other sum for the rolls from 1 to n and sum of 3 on the (n+1)th roll, still 3 before 7 - OK;
...

The above can be continued infinitely, and all above case represent the scenario when we have the sum of 3 BEFORE we have the sum of 7 (which will eventually occur on some roll afterwards). So the probability of getting 3 before 7 would be the sum of the probabilities of the above events.

Hope it's clear. Anyway: you won't need this for GMAT, so don't worry too much.
_________________
Intern
Joined: 15 Apr 2010
Posts: 45
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

07 Oct 2010, 03:28
Thanks Bunuel now I understand it. However I think you miss the scenario where sum=7 occurs before sum=3. For example, 1st time sum=7, 2nd time sum=3, so the probability must be 6/36 * 2/36. You assumed that sum 7 never happens before sum 3 so you take the probability of other sum (7/9) to calculate. Am I wrong?
Math Expert
Joined: 02 Sep 2009
Posts: 51160
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

07 Oct 2010, 04:38
catennacio wrote:
Thanks Bunuel now I understand it. However I think you miss the scenario where sum=7 occurs before sum=3. For example, 1st time sum=7, 2nd time sum=3, so the probability must be 6/36 * 2/36. You assumed that sum 7 never happens before sum 3 so you take the probability of other sum (7/9) to calculate. Am I wrong?

Yes, you are wrong. I'm not assuming that 7 doesn't occur before 3 rather than I'm only interested to count the probability of the cases when 3 occur before 7 as this is what the question is asking, that's why I'm not considering the cases when 7 occur before 3.
_________________
Manager
Joined: 27 May 2008
Posts: 104
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

07 Oct 2010, 22:40
Nice explanation Bunuel !
Intern
Joined: 15 Apr 2010
Posts: 45
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

08 Oct 2010, 18:11
Thanks Bunuel, you're right.. sometimes I got off the track when thinking... thanks for being patient with me
Non-Human User
Joined: 09 Sep 2013
Posts: 9143
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR  [#permalink]

Show Tags

03 Oct 2017, 12:36
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: You have two dice, what is the probability of rolling a sum of 3 BEFOR &nbs [#permalink] 03 Oct 2017, 12:36
Display posts from previous: Sort by