Probability and a Bag of Cards

Expert replies

dtweah: Master | Next Rank: 500 Posts; Posts: 487; Joined: Fri Mar 27, 2009 5:49 am; Thanked: 36 times

by dtweah » Sat May 02, 2009 9:55 am

A bag has cards with 3 questions written on each card. There are 20 cards in the bag, 8 of which are favorable ( you know the answer to all 3 questions on each of the favorable cards). If You answer all three questions you get an A. If cards are drawn at random from the bag, what is the probability you get an A on the second draw?

A. 1/4

B. 1/3

C. 1/5

D. 2/5

E. 1/2

Like this post

Source: Beat The GMAT — Problem Solving | Sat May 02, 2009 9:55 am

DeepakR: Master | Next Rank: 500 Posts; Posts: 158; Joined: Tue Sep 30, 2008 8:47 amFollowed by: 1 member; Thanked: 23 times; GMAT Score:660

by DeepakR » Sat May 02, 2009 10:23 am

We have Total 20 cards of which 8 are favourable hence 12 are unfavorable.

1st card is an unfavorable card = P(Unfav)=12C1/20C1=12/20
2nd card is favorable = P(Fav)=8C1/20C1=8/20.

Hence P=(12/20)*(8/20)=6/30=1/5 = C.) is it correct ?

-Deepak

Like this post

dumb.doofus: Master | Next Rank: 500 Posts; Posts: 435; Joined: Sat Sep 27, 2008 2:02 pmFollowed by: 1 member; Location: San Jose, CA; Thanked: 43 times; GMAT Score:720

by dumb.doofus » Sat May 02, 2009 12:40 pm

I think its D i.e. 2/5 Here is how I solved it..

We have to find probability of getting A on the second draw.. this leads us to two conditions
1. Getting A on both draws
Probability of that is = (8/20)*(7/19) --- (1)

2. Getting A only on 2nd draw
Probability of that is = (12/20)*(8/19) --- (2)

So total probability = (1) + (2)
= 152/380
= 2/5

One love, one blood, one life. You got to do what you should.
https://dreambigdreamhigh.blocked/
https://gmattoughies.blocked/

Like this post

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat May 02, 2009 2:52 pm

dumb.doofus wrote:I think its D i.e. 2/5 Here is how I solved it..

We have to find probability of getting A on the second draw.. this leads us to two conditions
1. Getting A on both draws
Probability of that is = (8/20)*(7/19) --- (1)

2. Getting A only on 2nd draw
Probability of that is = (12/20)*(8/19) --- (2)

So total probability = (1) + (2)
= 152/380
= 2/5

This strategy is 100% accurate. However, what would we do if the question asked us to find P(getting an A on 13th card). The calculations would get pretty complicated.

Notice that P(getting an A on 2nd card) = P(getting an A on 1st card)
We have 20 cards and 8 of them are "A" cards. So, P(getting an A on 1st card) = 8/20 = 2/5

The point I wish to make is that P(getting an A on 1st card) = P(getting an A on 2nd card) = P(getting an A on 3rd card) etc.

To illustrate my point, imagine the following scenario:
Someone shuffles the deck of cards and selects a card. Now you enter the room.
The person asks you, "What is the probability that this card is an A card?"
Does it matter if the card in question was the first card drawn? No.
The probability will be the same regardless of whether the card is the first drawn or the last drawn.

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

dumb.doofus: Master | Next Rank: 500 Posts; Posts: 435; Joined: Sat Sep 27, 2008 2:02 pmFollowed by: 1 member; Location: San Jose, CA; Thanked: 43 times; GMAT Score:720

by dumb.doofus » Sat May 02, 2009 3:03 pm

Brent Hanneson wrote:
dumb.doofus wrote:I think its D i.e. 2/5 Here is how I solved it..

We have to find probability of getting A on the second draw.. this leads us to two conditions
1. Getting A on both draws
Probability of that is = (8/20)*(7/19) --- (1)

2. Getting A only on 2nd draw
Probability of that is = (12/20)*(8/19) --- (2)

So total probability = (1) + (2)
= 152/380
= 2/5
This strategy is 100% accurate. However, what would we do if the question asked us to find P(getting an A on 13th card). The calculations would get pretty complicated.

Notice that P(getting an A on 2nd card) = P(getting an A on 1st card)
We have 20 cards and 8 of them are "A" cards. So, P(getting an A on 1st card) = 8/20 = 2/5

The point I wish to make is that P(getting an A on 1st card) = P(getting an A on 2nd card) = P(getting an A on 3rd card) etc.

To illustrate my point, imagine the following scenario:
Someone shuffles the deck of cards and selects a card. Now you enter the room.
The person asks you, "What is the probability that this card is an A card?"
Does it matter if the card in question was the first card drawn? No.
The probability will be the same regardless of whether the card is the first drawn or the last drawn.

Brent I have my doubts about that.. if I was replacing the card back in the deck then what you have mentioned would hold good. But if I am not replacing the card back in the deck, which is what I have done in my solution, then the probability will always change..

Taking your example, if I have to find the probability on the 13th card out of the 20 cards..
then as you can see.. if the first 12 were not A cards.. probability of getting an A card on the 13th try is 1, whereas it won't be 1 if in the 12 trys that I had, some times I did get an A card..

One love, one blood, one life. You got to do what you should.
https://dreambigdreamhigh.blocked/
https://gmattoughies.blocked/

Like this post

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat May 02, 2009 3:19 pm

Okay, let me ask a slightly different question.
Let's say only one of the 20 cards is an "A card" and we draw each card without replacement.

What is P(1st card drawn is A card)?
We get 1/20

What is P(2nd card drawn is A card)?
We get (19/20)(1/19)=1/20

What is P(3rd card drawn is A card)?
We get (19/20)(18/19)(1/18)=1/20

Notice that numerators and denominators keep cancelling out. If you try the same calculations on the original question, you will keep getting 2/5

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat May 02, 2009 3:23 pm

Here's a slightly different angle.
We have the original 20 cards and 8 of them are "A cards"
You enter the room and one card has been taken from the deck and is face down on the table.
Someone asks you to determine P(the card is an A card). Do we really need to know whether this card is the 1st, 2nd, 3rd or last card for us to be able to determine the probability?

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

dumb.doofus: Master | Next Rank: 500 Posts; Posts: 435; Joined: Sat Sep 27, 2008 2:02 pmFollowed by: 1 member; Location: San Jose, CA; Thanked: 43 times; GMAT Score:720

by dumb.doofus » Sat May 02, 2009 3:29 pm

Hmmmm.. I am getting your point Brent.. Awesome.. thanks so much.. you are absolutely right..

One love, one blood, one life. You got to do what you should.
https://dreambigdreamhigh.blocked/
https://gmattoughies.blocked/

Like this post

dtweah: Master | Next Rank: 500 Posts; Posts: 487; Joined: Fri Mar 27, 2009 5:49 am; Thanked: 36 times

by dtweah » Sat May 02, 2009 5:59 pm

dumb.doofus wrote:I think its D i.e. 2/5 Here is how I solved it..

We have to find probability of getting A on the second draw.. this leads us to two conditions
1. Getting A on both draws
Probability of that is = (8/20)*(7/19) --- (1)

2. Getting A only on 2nd draw
Probability of that is = (12/20)*(8/19) --- (2)

So total probability = (1) + (2)
= 152/380
= 2/5

Good Job Dumb. Keep nailing them probability. OA is 2/5.

Like this post

DeepakR: Master | Next Rank: 500 Posts; Posts: 158; Joined: Tue Sep 30, 2008 8:47 amFollowed by: 1 member; Thanked: 23 times; GMAT Score:660

by DeepakR » Sat May 02, 2009 6:17 pm

Thanks a lot Dumb. Your answers are very clear and helps a lot. I have a doubt in this question. If nothing is specified do we need to consider the question as without replacement. ?

Thanks Brent that was a great tip.

- Deepak

Like this post

dumb.doofus: Master | Next Rank: 500 Posts; Posts: 435; Joined: Sat Sep 27, 2008 2:02 pmFollowed by: 1 member; Location: San Jose, CA; Thanked: 43 times; GMAT Score:720

by dumb.doofus » Sun May 03, 2009 12:19 am

Thanks buddies..

Well, with or without replacement are kind of very clear from the words that are used in the question..

For ex: in this question, it says, cards are drawn at random from the bag.. essentially states that you are taking out cards from the bag and not keeping them back in..

Some questions very specifically mention with or without replacement.. and some do not.. but the wordings used in a question gives it out..

HTH..

One love, one blood, one life. You got to do what you should.
https://dreambigdreamhigh.blocked/
https://gmattoughies.blocked/

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review