Last visit was: 08 Jul 2025, 10:44 It is currently 08 Jul 2025, 10:44
Close
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
Your Progress

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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
505-555 Level|   Probability|                              
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 8 July 2025
Posts: 102,593
Own Kudos:
Given Kudos: 97,451
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,593
Kudos: 739,485
 [191]
27
Kudos
Add Kudos
163
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
GMATinsight
User avatar
Major Poster
Joined: 08 Jul 2010
Last visit: 08 Jul 2025
Posts: 6,371
Own Kudos:
Given Kudos: 128
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,371
Kudos: 15,559
 [101]
47
Kudos
Add Kudos
54
Bookmarks
Bookmark this Post
avatar
g3lo18
Joined: 31 Jan 2016
Last visit: 07 Mar 2019
Posts: 14
Own Kudos:
24
 [17]
Given Kudos: 6
Posts: 14
Kudos: 24
 [17]
10
Kudos
Add Kudos
6
Bookmarks
Bookmark this Post
General Discussion
User avatar
AkshdeepS
Joined: 13 Apr 2013
Last visit: 08 Jul 2025
Posts: 1,441
Own Kudos:
1,821
 [13]
Given Kudos: 1,001
Status:It's near - I can see.
Location: India
Concentration: International Business, Operations
GPA: 3.01
WE:Engineering (Real Estate)
Products:
Posts: 1,441
Kudos: 1,821
 [13]
9
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4


Kudos for a correct solution.


My Solution:

Total Pens = 12 Nos

Defective pens = 3

Remaining = 9, Therefore, selecting 2 no defective pens from 9 = 9C2 ways, and selecting 2 pens from 12 = 12C2 ways

Probability of neither pen will be defective = 9C2/12C2 = 36/66 = 6/11 Answer is C
User avatar
Skywalker18
User avatar
Retired Moderator
Joined: 08 Dec 2013
Last visit: 15 Nov 2023
Posts: 2,052
Own Kudos:
9,677
 [4]
Given Kudos: 171
Status:Greatness begins beyond your comfort zone
Location: India
Concentration: General Management, Strategy
GPA: 3.2
WE:Information Technology (Consulting)
Products:
Posts: 2,052
Kudos: 9,677
 [4]
2
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
Total number of pens = 12
Defective pens = 3
Non- defective pens= 9

Probablity of selecting 2 non defective pens = 9C2/ 12C2 = 6/11
Answer C

Alternatively , we can use probablity = No of favorable outcomes / No of total outcomes
= 9/12 * 8/11 = 6/11
User avatar
BillyZ
User avatar
Current Student
Joined: 14 Nov 2016
Last visit: 03 May 2025
Posts: 1,147
Own Kudos:
21,905
 [1]
Given Kudos: 926
Location: Malaysia
Concentration: General Management, Strategy
GMAT 1: 750 Q51 V40 (Online)
GPA: 3.53
Products:
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4


Kudos for a correct solution.

Bunuel What if Probability (Neither pen will be defective) = 1-Probability (defective both time) = 1- (3/12)*(2/11) = 21/22 ?

I get the different yield.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 8 July 2025
Posts: 102,593
Own Kudos:
739,485
 [6]
Given Kudos: 97,451
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,593
Kudos: 739,485
 [6]
2
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
ziyuenlau
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4


Kudos for a correct solution.
Bunuel What if Probability (Neither pen will be defective) = 1-Probability (defective both time) = 1- (3/12)*(2/11) = 21/22 ?

I get the different yield.
P(Neither pen will be defective) =

= 1 - (P(both pens are defective) + P(one of the pens is defective)) =

= 1 - (3/12*2/11 + 2*3/12*9/11) =

= 6/11.

Hope it's clear.­
avatar
elhho
Joined: 15 Aug 2016
Last visit: 11 Jun 2019
Posts: 2
Given Kudos: 3
Posts: 2
Kudos: 0
Kudos
Add Kudos
Bookmarks
Bookmark this Post
How do you know when to use the (9/12) * (8/11) and when to use (9/12) * (9/12)? It doesn't explicitly say that the customer picks one pen and then another pen afterward. Couldn't the customer have taken both pens at the same time?
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 8 July 2025
Posts: 102,593
Own Kudos:
739,485
 [1]
Given Kudos: 97,451
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,593
Kudos: 739,485
 [1]
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
elhho
How do you know when to use the (9/12) * (8/11) and when to use (9/12) * (9/12)? It doesn't explicitly say that the customer picks one pen and then another pen afterward. Couldn't the customer have taken both pens at the same time?

1. If the drawing is with replacement it's explicitly mentioned. If it's not mentioned, then it's without replacement.

2. Mathematically the probability of picking two balls simultaneously, or picking them one at a time (without replacement) is the same.
User avatar
JeffTargetTestPrep
User avatar
Target Test Prep Representative
Joined: 04 Mar 2011
Last visit: 05 Jan 2024
Posts: 2,997
Own Kudos:
7,911
 [2]
Given Kudos: 1,646
Status:Head GMAT Instructor
Affiliations: Target Test Prep
Expert
Expert reply
Posts: 2,997
Kudos: 7,911
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4

Since there are 3 defective pens from 12, the probability of selecting the first non-defective pen is 9/12 and the probability of selecting the second non-defective pen is 8/11. Thus, the probability that a customer buys 2 non-defective pens is 9/12 x 8/11 = 3/4 x 8/11 = 24/44 = 6/11.

Answer: C
User avatar
BrentGMATPrepNow
User avatar
Major Poster
Joined: 12 Sep 2015
Last visit: 13 May 2024
Posts: 6,756
Own Kudos:
34,031
 [1]
Given Kudos: 799
Location: Canada
Expert
Expert reply
Posts: 6,756
Kudos: 34,031
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4

P(neither pen is defective) = P(1st pen selected is NOT defective AND 2nd pen selected is NOT defective)
= P(1st pen selected is NOT defective) x P(2nd pen selected is NOT defective)
= 9/12 x 8/11
= 6/11
= C

ASIDE: How did I get 9/12 and 8/11?
For the first selection, 9 of the 12 pens are good.
For the second selection, we must assume that the first selection resulted in a GOOD pen. This means there are now 11 pens remaining, and 8 of them are GOOD.
User avatar
suganyam
Joined: 19 Jan 2019
Last visit: 01 Oct 2022
Posts: 40
Own Kudos:
Given Kudos: 65
Products:
Posts: 40
Kudos: 9
Kudos
Add Kudos
Bookmarks
Bookmark this Post
hi can some kindly point out my mstake
3c2/12c2+(3c1*9c1/12c2)2

I understand that it should be 3c1/12*9c1/11 but how should i distinguish them. like which one to use where Kindly guide me
Zillion thanks in advance­
User avatar
sujoykrdatta
Joined: 26 Jun 2014
Last visit: 07 Jul 2025
Posts: 539
Own Kudos:
1,038
 [2]
Given Kudos: 13
Status:Mentor & Coach | GMAT Q51 | CAT 99.98
Expert
Expert reply
Posts: 539
Kudos: 1,038
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
suganyam
hi can some kindly point out my mistake
3c2/12c2+(3c1*9c1/12c2)2

I understand that it should be 3c1/12*9c1/11 but how should i distinguish them. like which one to use where Kindly guide me
Zillion thanks in advance

Question:
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

There are 9 non-defective pens out of 12

If you want to pick simultaneously VS you pick one at a time without replacement, the answers would be SAME

If you want to do this by picking simultaneously, you should interpret the question as:
Favorable: Pick 2 pens that are not defective i.e. pick 2 pens from 9 = 9C2 ways = 9*8/2! = 36 ways
Total: Pick 2 pens from 12 = 12C2 ways = 12*11/2! = 66 ways
=> Probability = 36/66 = 6/11

If you want to do this by picking one at a time, you should interpret the question as:
Pick 1st pen which is not defective AND Pick 2nd pen which is not defective (Note: AND means MULTIPLY)

Probability of picking 1st pen which is not defective = 9/12

Not, since we pick WITHOUT REPLACEMENT, we have 8 non-defective out of 11

Thus, probability of picking 2nd pen which is not defective = 8/11

Thus, required probability = 9/12 * 8/11 = 6/11­
User avatar
ScottTargetTestPrep
User avatar
Target Test Prep Representative
Joined: 14 Oct 2015
Last visit: 08 Jul 2025
Posts: 21,053
Own Kudos:
Given Kudos: 296
Status:Founder & CEO
Affiliations: Target Test Prep
Location: United States (CA)
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 21,053
Kudos: 26,104
Kudos
Add Kudos
Bookmarks
Bookmark this Post
suganyam
hi can some kindly point out my mstake
3c2/12c2+(3c1*9c1/12c2)2


Your reasoning is not correct. Suppose you are choosing 2 pencils out of 9 pencils which are all good; i.e. there are no defective pencils in the box. The two pencils you choose will certainly be both non-defective. In this case, the probability that neither pencil is defective is 1 or equivalently, 100%. According to your reasoning, since there are 9 good pencils and we are choosing 2, the probability should have been 2/9.
Conversely, suppose that there are like a million defective pencils in the box and only 9 good pencils. Again according to your method, since there are 9 good pencils and we are choosing 2, the probability should have been 2/9 but in reality, you can observe that if there were a million defective pencils and only 9 good pencils, the probability of choosing two good pencils is very very low, almost zero.

For the other question you mentioned, if there were 5 red and 4 blue balls in a box and we were choosing 2 balls without replacement; the probability that both are red is (5/9)*(4/8) = 5/18 and the probability that both are blue is (4/9)*(3/8) = 1/6.
User avatar
GMATinsight
User avatar
Major Poster
Joined: 08 Jul 2010
Last visit: 08 Jul 2025
Posts: 6,371
Own Kudos:
Given Kudos: 128
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,371
Kudos: 15,559
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4


Answer: Option C

Step-by-Step Video solution by GMATinsight

User avatar
CrackverbalGMAT
User avatar
Major Poster
Joined: 03 Oct 2013
Last visit: 08 Jul 2025
Posts: 4,847
Own Kudos:
Given Kudos: 225
Affiliations: CrackVerbal
Location: India
Expert
Expert reply
Posts: 4,847
Kudos: 8,619
Kudos
Add Kudos
Bookmarks
Bookmark this Post
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

In a box of 12 pens, 9 of them are not defective and 3 are defective. We are selecting 2 pens at random.

Total outcomes here is selecting 2 pens out of 12 = 12C2
Favorable outcomes = Selecting 2 pens out of 9 non defective pens = 9 C2

P(neither pen will be defective) = Fav Outcomes/ Total Outcomes = 9C2/12C2 = 9*8/12*11 = 6/11

Option C is the answer.

Thanks,
Clifin J Francis,
GMAT SME
User avatar
PGTLrowanhand
Joined: 30 Oct 2012
Last visit: 18 Jun 2025
Posts: 75
Own Kudos:
167
 [1]
Given Kudos: 3
Status:London UK GMAT Consultant / Tutor
Expert
Expert reply
Posts: 75
Kudos: 167
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi GMATters,

Here's my video solution to this problem:
User avatar
shwetasood
Joined: 04 Jun 2023
Last visit: 06 Jun 2024
Posts: 48
Own Kudos:
Given Kudos: 280
Posts: 48
Kudos: 20
Kudos
Add Kudos
Bookmarks
Bookmark this Post
JeffTargetTestPrep
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4
Since there are 3 defective pens from 12, the probability of selecting the first non-defective pen is 9/12 and the probability of selecting the second non-defective pen is 8/11. Thus, the probability that a customer buys 2 non-defective pens is 9/12 x 8/11 = 3/4 x 8/11 = 24/44 = 6/11.

Answer: C
­
 Why are we multiplying? The question says "neither" of the pens so its using the "OR" condition and so shouldn't we add the probabilities?
 
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 8 July 2025
Posts: 102,593
Own Kudos:
Given Kudos: 97,451
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,593
Kudos: 739,485
Kudos
Add Kudos
Bookmarks
Bookmark this Post
shwetasood
JeffTargetTestPrep
Bunuel
In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective?

A. 1/6
B. 2/9
C. 6/11
D. 9/16
E. 3/4
Since there are 3 defective pens from 12, the probability of selecting the first non-defective pen is 9/12 and the probability of selecting the second non-defective pen is 8/11. Thus, the probability that a customer buys 2 non-defective pens is 9/12 x 8/11 = 3/4 x 8/11 = 24/44 = 6/11.

Answer: C
­
 Why are we multiplying? The question says "neither" of the pens so its using the "OR" condition and so shouldn't we add the probabilities?

 

­"Neither pen will be defective" means that the first pen is not defective AND the second pen is not defective. Therefore, we multiply the probabilities of each event to find the probability that both conditions are satisfied simultaneously.
User avatar
DanTheGMATMan
Joined: 02 Oct 2015
Last visit: 08 Jul 2025
Posts: 353
Own Kudos:
Given Kudos: 9
Expert
Expert reply
Posts: 353
Kudos: 168
Kudos
Add Kudos
Bookmarks
Bookmark this Post
­Classic probability:

 1   2   
Moderators:
Math Expert
102593 posts
PS Forum Moderator
678 posts