BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Probability

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Legendary Member
Posts: 641
Joined: Tue Feb 14, 2012 3:52 pm
Thanked: 11 times
Followed by:8 members

Probability

by gmattesttaker2 » Sat Sep 08, 2012 2:19 am
Hello,

This is from MGAT Guide 5 P. 65. Can you please tell me if my approach is correct? Also, can this problem also be solved using the slot method? Thanks for your help. Best Regards - Sri


A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?

Book answer:[spoiler] 40%[/spoiler]

My approach:
(3/5).(2/4) + (2/5).(1/4) = 8/20
Top
Source: — Problem Solving |
Senior | Next Rank: 100 Posts
Posts: 58
Joined: Mon Aug 27, 2012 3:09 am
Thanked: 2 times

by akashkumar1987 » Sat Sep 08, 2012 3:10 am
Hi,

By 40% do u mean the answer s 4.If yes then explanation is below

No of selections that he will do in total = 5C2 = 10 i.e he has to select 2 animals out of 5.

Now u want both the same so to select diff animals = 3*2= 6

Finally we want the matched pair = Total ways of selecting - Ways of selecting diff animal
= 10 - 6
= 4.

Thanks.
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Sat Sep 08, 2012 6:54 am
gmattesttaker2 wrote:Hello,

This is from MGAT Guide 5 P. 65. Can you please tell me if my approach is correct? Also, can this problem also be solved using the slot method? Thanks for your help. Best Regards - Sri


A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?

Book answer:[spoiler] 40%[/spoiler]

My approach:
(3/5).(2/4) + (2/5).(1/4) = 8/20
Your approach looks great!

For those who aren't quite sure how you arrived at this, I'll lengthen the solution.

First notice that, to get a matched pair, we can select 2 doves or 2 rabbits.

So, P(matched pair) = P(1st pick is rabbit AND 2nd pick is rabbit OR 1st pick is dove AND 2nd pick is dove)

We can now apply our AND and OR rules to get:
P(matched pair) = [P(1st pick is rabbit) X P(2nd pick is rabbit)] + [P(1st pick is dove) X P(2nd pick is dove)]
So, P(matched pair) = [(3/5) X (2/4)] + [(2/5) X (1/4)]
We get: 2/5 (or 0.4)

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Sat Sep 08, 2012 7:07 am
gmattesttaker2 wrote: A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?
We can apply the slot method here.
To begin, P(matched pair) = (# of ways to get a matched pair)/(# of ways to select 2 animals)

As always, begin with the denominator.

# of ways to select 2 animals
To count this, we'll treat each animal as different.
We'll take the task of selecting 2 animals and break it into stages.
Stage 1: Select the 1st animal. There are 5 animals, so this stage can be accomplished in 5 ways.
Stage 2: Select the 2nd animal. There are now 4 animals remaining, so this stage can be accomplished in 4 ways.
So, the total number of ways to select 2 animals is (5)(4), which equals 20

Now the numerator.

# of ways to get a matched pair
We need to consider two cases.

Case 1: select 2 doves.
In how many different ways can this occur?
Well, we'll take the task of selecting 2 doves and break it into stages.
Stage 1: Select the 1st dove. There are 3 doves, so this stage can be accomplished in 3 ways.
Stage 2: Select the 2nd dove. There are now 2 doves remaining, so this stage can be accomplished in 2 ways.
So, the total number of ways to select 2 doves is (3)(2), which equals 6

Case 2: select 2 rabbits.
In how many different ways can this occur?
Well, we'll take the task of selecting 2 rabbits and break it into stages.
Stage 1: Select the 1st rabbit. There are 2 rabbits, so this stage can be accomplished in 2 ways.
Stage 2: Select the 2nd rabbit. There is now 1 rabbit remaining, so this stage can be accomplished in 1 ways.
So, the total number of ways to select 2 rabbits is (2)(1), which equals 2

Put it all together to get:
P(matched pair) = (6+2)/(20)
= 8/20
= 2/5 (or 0.4)

Cheers,
Brent

Aside: The above solution uses something called the Fundamental Counting Principle (FCP). For more information about the FCP, we have a free video on the subject: https://www.gmatprepnow.com/module/gmat-counting?id=775
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
Legendary Member
Posts: 641
Joined: Tue Feb 14, 2012 3:52 pm
Thanked: 11 times
Followed by:8 members

by gmattesttaker2 » Sat Sep 08, 2012 11:57 pm
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote: A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?
We can apply the slot method here.
To begin, P(matched pair) = (# of ways to get a matched pair)/(# of ways to select 2 animals)

As always, begin with the denominator.

# of ways to select 2 animals
To count this, we'll treat each animal as different.
We'll take the task of selecting 2 animals and break it into stages.
Stage 1: Select the 1st animal. There are 5 animals, so this stage can be accomplished in 5 ways.
Stage 2: Select the 2nd animal. There are now 4 animals remaining, so this stage can be accomplished in 4 ways.
So, the total number of ways to select 2 animals is (5)(4), which equals 20

Now the numerator.

# of ways to get a matched pair
We need to consider two cases.

Case 1: select 2 doves.
In how many different ways can this occur?
Well, we'll take the task of selecting 2 doves and break it into stages.
Stage 1: Select the 1st dove. There are 3 doves, so this stage can be accomplished in 3 ways.
Stage 2: Select the 2nd dove. There are now 2 doves remaining, so this stage can be accomplished in 2 ways.
So, the total number of ways to select 2 doves is (3)(2), which equals 6

Case 2: select 2 rabbits.
In how many different ways can this occur?
Well, we'll take the task of selecting 2 rabbits and break it into stages.
Stage 1: Select the 1st rabbit. There are 2 rabbits, so this stage can be accomplished in 2 ways.
Stage 2: Select the 2nd rabbit. There is now 1 rabbit remaining, so this stage can be accomplished in 1 ways.
So, the total number of ways to select 2 rabbits is (2)(1), which equals 2

Put it all together to get:
P(matched pair) = (6+2)/(20)
= 8/20
= 2/5 (or 0.4)

Cheers,
Brent

Aside: The above solution uses something called the Fundamental Counting Principle (FCP). For more information about the FCP, we have a free video on the subject: https://www.gmatprepnow.com/module/gmat-counting?id=775
Hello Brent,

Hope all is well. Thanks a lot for the detailed and excellent explanation.

Best Regards,
Sri
Top
Legendary Member
Posts: 641
Joined: Tue Feb 14, 2012 3:52 pm
Thanked: 11 times
Followed by:8 members

by gmattesttaker2 » Thu Feb 28, 2013 12:18 am
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote:Hello,

This is from MGAT Guide 5 P. 65. Can you please tell me if my approach is correct? Also, can this problem also be solved using the slot method? Thanks for your help. Best Regards - Sri


A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?

Book answer:[spoiler] 40%[/spoiler]

My approach:
(3/5).(2/4) + (2/5).(1/4) = 8/20
Your approach looks great!

For those who aren't quite sure how you arrived at this, I'll lengthen the solution.

First notice that, to get a matched pair, we can select 2 doves or 2 rabbits.

So, P(matched pair) = P(1st pick is rabbit AND 2nd pick is rabbit OR 1st pick is dove AND 2nd pick is dove)

We can now apply our AND and OR rules to get:
P(matched pair) = [P(1st pick is rabbit) X P(2nd pick is rabbit)] + [P(1st pick is dove) X P(2nd pick is dove)]
So, P(matched pair) = [(3/5) X (2/4)] + [(2/5) X (1/4)]
We get: 2/5 (or 0.4)

Cheers,
Brent
Hello Brent,

Hope all is well and thanks for your detailed explanation above. When I tried solving this problem again I was thinking of the following:

2 doves can be selected out of 3 doves (D1,D2 and D3) as follows:
D1 D2
D1 D3
D2 D1
D2 D3
D3 D1
D3 D2

However, since D2 D1 is the same as D1 D2, D3 D1 is the same as D1 D3 and D3 D2 is the same as D2 D3 I divided (3/5)(2/4) by 2!.

Similarly for 2 rabbits, it would be (2/5).(1/4) divided by 2!

Hence, Probability of a matched pair = ( (3/5)(2/4) )/2! + ( (2/5)(1/4) )/2!

However, I don't think this approach is correct. I was just wondering if you can please assist? Thanks for all your help.

Best Regards,
Sri
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Thu Feb 28, 2013 7:47 am
gmattesttaker2 wrote: Hello Brent,

Hope all is well and thanks for your detailed explanation above. When I tried solving this problem again I was thinking of the following:

2 doves can be selected out of 3 doves (D1,D2 and D3) as follows:
D1 D2
D1 D3
D2 D1
D2 D3
D3 D1
D3 D2

However, since D2 D1 is the same as D1 D2, D3 D1 is the same as D1 D3 and D3 D2 is the same as D2 D3 I divided (3/5)(2/4) by 2!.

Similarly for 2 rabbits, it would be (2/5).(1/4) divided by 2!

Hence, Probability of a matched pair = ( (3/5)(2/4) )/2! + ( (2/5)(1/4) )/2!

However, I don't think this approach is correct. I was just wondering if you can please assist? Thanks for all your help.

Best Regards,
Sri
Hmmm, you seem to be combining counting techniques and probability rules. I suggest that you use only one approach.

Let's use each approach to find the probability of selecting 2 doves.

Probability Rules Approach:
So, P(2 doves) = P(1st pick is dove AND 2nd pick is dove)
= P(1st pick is dove) X P(2nd pick is dove)]
= (3/5) X (2/4)
= 3/10

Aside: I noticed that in my earlier posts, I mixed up doves and rabbits. My bad (the final probability is still the same though)


Counting Approach:
P(2 doves) = (# of outcomes in which 2 doves are selected)/(total # of outcomes)

# of outcomes in which 2 doves are selected:
There are 3 doves, and we must select 2.
Since the order of the selected doves does not matter, we can use combinations.
We can select 2 doves from 3 doves in 3C2 ways (= 3 ways)

Aside: If anyone is interested, we have a free video on calculating combinations (like 3C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

total # of outcomes
There are 5 animals, and we must select 2.
Since the order of the selected animals does not matter, we can use combinations.
We can select 2 animals from 5 animals in 5C2 ways (= 10 ways)

So, P(2 doves) = 3/10

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
Legendary Member
Posts: 512
Joined: Mon Jun 18, 2012 11:31 pm
Thanked: 42 times
Followed by:20 members

by sana.noor » Sat Mar 02, 2013 5:17 am
its an easy one
Probability formula = number of desired outcomes/ total possible outcomes
Number of desired outcomes = D1 D2, D1 D3 and D2 D3----> three pairs of dove
R1 R2----> one pair of rabbit as the bag has only two rabbits....thus the desired pairs will be equla to 4.
Total possible outcomes= apply formula of combinatorics = 5!/3! 2! = 10
putting this in formula = 4/10 = 40%
Work hard in Silence, Let Success make the noise.

If you found my Post really helpful, then don't forget to click the Thank/follow me button. :)
Top
User avatar
Newbie | Next Rank: 10 Posts
Posts: 7
Joined: Sun Jul 19, 2015 7:10 am

by alexibiz » Sun Jul 19, 2015 7:23 am
sana.noor wrote:its an easy one
Probability formula = number of desired outcomes/ total possible outcomes
Number of desired outcomes = D1 D2, D1 D3 and D2 D3----> three pairs of dove
R1 R2----> one pair of rabbit as the bag has only two rabbits....thus the desired pairs will be equla to 4.
Total possible outcomes= apply formula of combinatorics = 5!/3! 2! = 10
putting this in formula = 4/10 = 40%
Actually this approach seems to be faulty. I do agree with the other way of calculating the probability for this problem, but this one (the same as what manhatten provided) doesn't make sesnse!

d1d2, d1d3 and d2d3 do not make sense! You have 3!x2! in denominator due to same letter combinations that we do not need to count! What makes sense is the whole list of possible combinations (10). We can't look at this problem from the perspective of only two pulls, otherwise the possible number of outcomes becomes 4, not 10: dd, rd, dr, rr. That's it!

But here is what 10-combo (5!/3!x2!) means:

dddrr
ddrrd
drrdd
rrddd
rdrdd
drdrd
ddrdr
rdddr
rddrd
drddr

It just happened to be that we are only interested in the first two pulls. And from this table you can see that there are only 4 entries out of 10 start with two identical letters. Therefore, the answer to this problem is 40%.

So, the approach manhattanprep provided, at least in the way they presented it, is bogus. Am I correct?
Top
Legendary Member
Posts: 518
Joined: Tue May 12, 2015 8:25 pm
Thanked: 10 times

by nikhilgmat31 » Wed Jul 29, 2015 5:16 am
Matched Pair
d1,d2
d2,d3
d3,d1
r1,r2

total of 4 ways of matched pairs.

now denominator
to select 2 out 5 animals 5C2 i.e. 10

probability = 4/10 = 2/5
Top
Post new topic Post Reply
• Page 1 of 1