Last visit was: 19 Nov 2025, 03:20 It is currently 19 Nov 2025, 03:20
Home
GMAT
Messages
Tests
Schools
Events
Rewards
Chat
My Profile
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
605-655 Level|   Probability|      
User avatar
ctrlaltdel
User avatar
Joined: 24 Jul 2009
Last visit: 11 Jan 2023
Posts: 59
Own Kudos:
583
 [99]
Given Kudos: 124
Location: United States
GMAT 1: 590 Q48 V24
GMAT 1: 590 Q48 V24
Posts: 59
Kudos: 583
 [99]
Laura has a deck of standard playing cards with 13 of the 52 Updated on: 28 Nov 2012, 03:27
Originally posted by ctrlaltdel on 15 Nov 2009, 22:28.
Last edited by Bunuel on 28 Nov 2012, 03:27, edited 1 time in total.
Renamed the topic and edited the question.
9
Kudos
Add Kudos
89
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

45% (medium)

Question Stats:

65% (01:23) correct 35% (02:06) wrong based on 1013 sessions

History

Date
Time
Result
Not Attempted Yet
Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart." If Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?

(A) 1/4
(B) 1/5
(C) 5/26
(D) 12/42
(E) 13/42
ShowHide Answer
Official Answer
A
Most Helpful Reply
User avatar
IanStewart
User avatar
GMAT Tutor
Joined: 24 Jun 2008
Last visit: 18 Nov 2025
Posts: 4,145
Own Kudos:
10,986
 [46]
Given Kudos: 99
Expert
Expert reply
Posts: 4,145
Kudos: 10,986
 [46]
Laura has a deck of standard playing cards with 13 of the 52 16 Nov 2009, 03:13
34
Kudos
Add Kudos
12
Bookmarks
Bookmark this Post
Every card has a 13/52 = 1/4 chance of being a heart; it doesn't matter if it's the top card in the deck or the tenth card in the deck.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 Nov 2025
Posts: 105,379
Own Kudos:
778,190
 [19]
Given Kudos: 99,977
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 105,379
Kudos: 778,190
 [19]
Laura has a deck of standard playing cards with 13 of the 52 28 Nov 2012, 04:11
2
Kudos
Add Kudos
17
Bookmarks
Bookmark this Post
himanshuhpr
and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?



WHAT does this statement implies that 10 cards are withdrawn with replacement or without replacement ??
Show more

No replacement there, 10 cards are dealt and we are asked to find the probability that 10th card is a heart.

Similar questions to practice:
a-certain-team-has-12-members-including-joey-a-three-131321.html
a-certain-television-show-has-15-sponsors-including-company-127423.html
a-medical-researcher-must-choose-one-of-14-patients-to-127396.html
a-certain-class-consists-of-8-students-including-kim-127730.html
a-certain-club-has-10-members-including-harry-one-of-the-134891.html
General Discussion
User avatar
himanshuhpr
User avatar
Joined: 29 Aug 2012
Last visit: 28 Nov 2014
Posts: 24
Own Kudos:
193
 [1]
Given Kudos: 56
Schools: Babson '14
GMAT Date: 02-28-2013
Schools: Babson '14
Posts: 24
Kudos: 193
 [1]
Laura has a deck of standard playing cards with 13 of the 52 27 Nov 2012, 14:05
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?



WHAT does this statement implies that 10 cards are withdrawn with replacement or without replacement ??
User avatar
himanshuhpr
User avatar
Joined: 29 Aug 2012
Last visit: 28 Nov 2014
Posts: 24
Own Kudos:
193
 [3]
Given Kudos: 56
Schools: Babson '14
GMAT Date: 02-28-2013
Schools: Babson '14
Posts: 24
Kudos: 193
 [3]
Laura has a deck of standard playing cards with 13 of the 52 28 Nov 2012, 08:30
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
himanshuhpr
and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?



WHAT does this statement implies that 10 cards are withdrawn with replacement or without replacement ??

No replacement there, 10 cards are dealt and we are asked to find the probability that 10th card is a heart.

Show more

If there is no replacement then how is the (P) that the 10th card is 13/52 ??

there are many cases here to be considered here if there is no replacement such as:

H- Denotes heart X-may be any diamond, spade or club.

1. HXXXXXXXXH
2. HHXXXXXXXH
3. HHHXXXXXXH
.
.
.
.
.
9. HHHHHHHHHH
10. XXXXXXXXXH

All cases from 1 to 10 will have different probabilities for heart to be at the 10th place and it will take hell lot of time to calculate all of them.

For according to me the above solution by Ian is only valid if cards are replaced (Every card has a 13/52 = 1/4 chance of being a heart; it doesn't matter if it's the top card in the deck or the tenth card in the deck.)If that's the case that brings back me to my original question ---- how do we determine that the cards are replaced or not ?? based on the question given ....
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 Nov 2025
Posts: 105,379
Own Kudos:
778,190
 [1]
Given Kudos: 99,977
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 105,379
Kudos: 778,190
 [1]
Laura has a deck of standard playing cards with 13 of the 52 29 Nov 2012, 04:51
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
himanshuhpr
Bunuel
himanshuhpr
and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?



WHAT does this statement implies that 10 cards are withdrawn with replacement or without replacement ??

No replacement there, 10 cards are dealt and we are asked to find the probability that 10th card is a heart.


If there is no replacement then how is the (P) that the 10th card is 13/52 ??

there are many cases here to be considered here if there is no replacement such as:

H- Denotes heart X-may be any diamond, spade or club.

1. HXXXXXXXXH
2. HHXXXXXXXH
3. HHHXXXXXXH
.
.
.
.
.
9. HHHHHHHHHH
10. XXXXXXXXXH

All cases from 1 to 10 will have different probabilities for heart to be at the 10th place and it will take hell lot of time to calculate all of them.

For according to me the above solution by Ian is only valid if cards are replaced (Every card has a 13/52 = 1/4 chance of being a heart; it doesn't matter if it's the top card in the deck or the tenth card in the deck.)If that's the case that brings back me to my original question ---- how do we determine that the cards are replaced or not ?? based on the question given ....
Show more

When we have a case with replacement it's always clearly mentioned in the question. We are told that "Laura deals 10 cards off the top of the deck", which means that there is no replacement whatsoever.

As for the question, concept behind it is discussed in the topics given in my previous post.
avatar
jns
avatar
Joined: 21 Jun 2013
Last visit: 19 Jan 2018
Posts: 25
Own Kudos:
78
 [3]
Given Kudos: 129
Posts: 25
Kudos: 78
 [3]
Laura has a deck of standard playing cards with 13 of the 52 22 Sep 2013, 10:52
1
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
ctrlaltdel
Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart." If Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?
Show more

Hi! I went through all the similar problems provided by Bunuel and was able to understand and solve them. However, I am still not able to get why this is not an arrangement problem. Why do we need to ignore the number 10? Bunuel/Karishma, kindly elaborate.

I solved the question using reverse probability: 1- P(10th is not heart) = 1- (3*13)/52 = 1/4.
User avatar
Timebomb
User avatar
Joined: 29 Nov 2016
Last visit: 07 Oct 2020
Posts: 188
Own Kudos:
66
 [8]
Given Kudos: 446
Location: India
Schools: Stern '22 (A) Booth '22 (D) CBS '22 (D) Wharton '22 (D) Duke '22 (D) Kellogg '22 (A) Tuck '22 (D) Darden '22 (D)
GMAT 1: 750 Q50 V42
Schools: Stern '22 (A) Booth '22 (D) CBS '22 (D) Wharton '22 (D) Duke '22 (D) Kellogg '22 (A) Tuck '22 (D) Darden '22 (D)
GMAT 1: 750 Q50 V42
Posts: 188
Kudos: 66
 [8]
Laura has a deck of standard playing cards with 13 of the 52 18 Feb 2018, 23:07
5
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
I was confused as well,but the solution is something like this.
Let's assume the question is to find the probability of drawing second card as heart instead of 10th?

How we will approach this problem is:
A) find the probability that first drawn Is heart and second drawn is heart too i.e. (13/52*12/51)
B) find the probability that first drawn is not a heart and second drawn is a heart(39/52*13/51)

Adding the two will give you 1/4.

The above example is to give a sense of what is happening in this question.

Since we don't know the outcome we can't assume weather the first draw is a heart or not a heart. We added the outcome of two situations.

I see a lot of people saying how can the probability of drawing 10th card as a heart be 1/4 when all the cards from 1 to 9 can be heart. Yes! Certainly it can be but its one of the cases. Adding all such cases will give you 1/4.

And had we known the result of previous outcomes, indeed the question would be different.

As Bunuel said. May be laura knows the outcome but we don't.

Great question.
Thanks guys!

Regards
M

Posted from my mobile device

Posted from my mobile device
User avatar
IanStewart
User avatar
GMAT Tutor
Joined: 24 Jun 2008
Last visit: 18 Nov 2025
Posts: 4,145
Own Kudos:
10,986
 [18]
Given Kudos: 99
Expert
Expert reply
Posts: 4,145
Kudos: 10,986
 [18]
Laura has a deck of standard playing cards with 13 of the 52 07 May 2019, 08:42
14
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
For anyone looking at this problem and thinking "I need to consider what the first 9 cards are", perhaps I can offer an easy explanation of why that's unnecessary (if you did not think that, there's no reason to read any further!).

In a deck of cards, we have 13 hearts, 13 spades, 13 clubs and 13 diamonds, out of 52 cards, so 1/4 of the deck is hearts. Consider these questions:

If a magician spreads out a deck of cards, and asks you to choose one at random, what's the probability you pick a heart? It's 13/52 = 1/4, since you're just picking one card at random.

Now if the magician fans the deck of cards, and asks you to choose one, and you happen to pick the 10th card, what is the probability you picked a heart? Again, it's 1/4, for the same reason as before; you're still picking a random card from the deck. Whether you pick the 1st card, the 10th card, the 37th card or the last card, each has a 1/4 probability of being a heart.

But that's the same question as the one in the OP above - Laura is just picking the tenth card from the deck. Unless you have some information about the top nine cards, the tenth card is just as random as the first one.
avatar
PRAJESH0133
avatar
Joined: 18 Jul 2020
Last visit: 17 Mar 2024
Posts: 4
Own Kudos:
1
Given Kudos: 7
Posts: 4
Kudos: 1
Laura has a deck of standard playing cards with 13 of the 52 19 Nov 2020, 10:57
Kudos
Add Kudos
Bookmarks
Bookmark this Post
My question may sound incorrect. Why can't I think of the problem as geometric distribution in terms of first success ( heart) in k (10) trials. Please tell where I am going wrong

Posted from my mobile device
User avatar
IanStewart
User avatar
GMAT Tutor
Joined: 24 Jun 2008
Last visit: 18 Nov 2025
Posts: 4,145
Own Kudos:
10,986
 [1]
Given Kudos: 99
Expert
Expert reply
Posts: 4,145
Kudos: 10,986
 [1]
Laura has a deck of standard playing cards with 13 of the 52 19 Nov 2020, 11:15
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
PRAJESH0133
My question may sound incorrect. Why can't I think of the problem as geometric distribution in terms of first success ( heart) in k (10) trials. Please tell where I am going wrong

Posted from my mobile device
Show more

You don't need to know what a "geometric distribution" is for the GMAT, but I think you're trying to answer a more complicated question than the one asked. If you're thinking in terms of geometric distributions here, you're trying to answer a question like "what is the probability, if Laura picks cards one by one from the top of the deck, that she doesn't get a heart until she picks the tenth card?" The problem here is simpler - we don't need to even consider the first nine cards. It's just asking for the probability the tenth card is a heart, and nothing more.
User avatar
ParitoshRawat99
User avatar
Joined: 26 Nov 2020
Last visit: 06 Jul 2022
Posts: 116
Own Kudos:
37
Given Kudos: 82
Location: India
GMAT 1: 700 Q50 V35
GPA: 3.9
GMAT 1: 700 Q50 V35
Posts: 116
Kudos: 37
Laura has a deck of standard playing cards with 13 of the 52 01 May 2021, 03:32
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The answer is 1/4 Option A

When we shuffle and deal the 19 cards each card getting chosen would have a probability of 1/52.

And each card would have a probability of 1/4 being a heart.

The first 9 cards can be of any suite this probability = 1
The probability of the tenth card being a heart = 1/4

Thus Probability of 10th card being a heart = 1*1/4

Posted from my mobile device
User avatar
tanvi471
User avatar
Joined: 20 Dec 2020
Last visit: 21 Jun 2023
Posts: 15
Own Kudos:
3
Given Kudos: 141
Location: India
Schools: Ross '25 (II) Haas '25 (S) Fuqua '25 (S) IESE '25 Stern '25 (S)
Products:
My Reviews
Schools: Ross '25 (II) Haas '25 (S) Fuqua '25 (S) IESE '25 Stern '25 (S)
Posts: 15
Kudos: 3
Laura has a deck of standard playing cards with 13 of the 52 15 Jul 2022, 03:35
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Even I got it qrong first, but it's important to note that question doesn't mention about whethere first, second, third... etc was a heart or not. So replacement should not be taken into consideration.
So simply, withdrawal of a card at any point with hearts = 13/52
User avatar
Risdeeg123
User avatar
Joined: 21 Apr 2020
Last visit: 24 Jun 2025
Posts: 1
Given Kudos: 4
Posts: 1
Kudos: 0
Laura has a deck of standard playing cards with 13 of the 52 16 Apr 2025, 05:52
Kudos
Add Kudos
Bookmarks
Bookmark this Post
An easier way to solve this:
Consider,
Possible arrangements of all 52 cards (if all are drawn one by one) = 52!
Now, ways of selecting a heart from the 13 available hearts (for the 10th position) = C(13,1) = 13
Possible arrangements of the remaining 51 cards in the remaining 51 positions: 51!

Probability (10th card is heart) = (13 x 51!)/52! = 13/52 = 1/4
Moderators:
Bunuel
Math Expert
105379 posts
Krunaal
Tuck School Moderator
805 posts