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A magician has five animals in his magic hat: 3 doves and 2

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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
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xALIx 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?

A. 2/5
B. 3/5
C. 1/5
D. 1/2
E. 7/5

We can also solve the question using counting methods

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
= A

Cheers,
Brent
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Re: a magician has five animals - probability [#permalink]
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3-doves, 2-rabbits

Probability that he draws 2 doves in the two turns = 3/5 * 1/3 + 2/4 * 1/2 = 9/20

Probability that he draws 2 rabbits in the two turns = 2/5 * 1/2 + 1/4 = 9/20

Total probability = 18/20

Am I totally off-course or ???
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Re: a magician has five animals - probability [#permalink]
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The OA is 40%
Good job buff, right on, better explanation than the MGMAT book.

Any other approaches are much welcome

The OE:
5c2 for 10 possible pairs, which = 10
then list the pairs in which the animals will match. Let's represent the rabbits with the letters A and B, and the doves with letters X, Y, and Z

Matched Pairs:
RaRb
DxDy
DxDz
DyDz

Therefore, the probability that the magician will randomly draw a matched set is 4/10 = 40%
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A magician has 5 animals in this magic hat: 3 doves and 2 [#permalink]
A magician has 5 animals in this magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, which is the chance that he will have a matched pair.

P( D 1st time) and P (D 2nd time) = 3/5 x 2/4 = 3/10
or
P (R 1st time) and P (R 2nd time) 1/5 x 1/4 = 1/20

3/10 + 1/20 = 7/20

However this is wrong

The right answer is 40%, could somebody tell me why my answer is wrong please

Thank you!

Originally posted by chibimoon on 07 Oct 2012, 22:38.
Last edited by Bunuel on 08 Oct 2012, 01:29, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Probability + Combinatorics: very confusing [#permalink]
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chibimoon wrote:
Hi all
Q. A magician has 5 animals in this magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, which is the chance that he will have a matched pair.

P( D 1st time) and P (D 2nd time) = 3/5 x 2/4 = 3/10
or
P (R 1st time) and P (R 2nd time) 1/5 x 1/4 = 1/20

3/10 + 1/20 = 7/20

However this is wrong

The right answer is 40%, could somebody tell me why my answer is wrong please

Thank you!

You've worked out the probability for doves correctly. Although for the rabbits, it should be 2/5 x 1/4 = 1/10

So we would get 3/10 + 1/10 = 4/10 = 40%
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Re: Probability + Combinatorics: very confusing [#permalink]
Thank you so much, that was just a very embarrassing mistake
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Re: a magician has five animals - probability [#permalink]
x-ALI-x 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?

P(both doves) + P(both rabbits) = p(matched pair)
(3/5)*(2/4) + (2/5)*(1/4) = 2/5

Hi How did you get the probability 2/4 and 1/4???

Thanks
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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
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(3C2+2C2)/5C2=2/5
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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
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xALIx 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?

A. 2/5
B. 3/5
C. 1/5
D. 1/2
E. 7/5

Probability of getting a pair: Dove (Dove) + Rabbit (Rabbit) = (3/5) (2/4) + (2/5)(1/4) = (6 + 2)/20 = (8)/20 = 2/5
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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
Can we interpret in the following way?

It shouldnt matter whichever animal he takes out in the first time - implies probability is 1.

What matters really is the second animal that he chooses.

As he has already chose 1 animal, so the remaining animals are 4. Hence whichever animal he pulls out will make a matched pair with the first chosen animal. Therefore the probability should be 1/4 or 25%

Whats wrong with this approach?
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A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
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xALIx 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?

A. 2/5
B. 3/5
C. 1/5
D. 1/2
E. 7/5

Hello There,
Number of total outcomes = n(S) = 5C2
Number of possible outcomes to get a matched pair = n(E) = Possible number of Combinations to get a pair from 3 Doves + Possible number of combinations to get a pair from 2 Rabbits
= 3C2 + 2C2
Probability of getting a matched pair = n(S)/n(E)
= 5C2/(3C2 + 2C2) = 2/5

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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
Hey there, here's an alternativ approach with combinatorics

Possible outcomes: we have repating elements here 3 Doves and 2 Rabbits --> 5!/(3!2!) = 10 outcomes
Desired outcomes: so we have two groups here, so a matched pair od Doves OR of rabitts:
Doves --> if 2 Doves are in the group 3!/(2!*1!) = 3, sam efor rabbits 2!/2!=1, so 3+1=4
Finally we have 4/10 = 2/5 (A)
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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
3C2/5C2 + 2C2/5C2 = 3/10 + 1/10 = 4/10 = 2/5. Hence (A).
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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
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xALIx 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?

A. 2/5
B. 3/5
C. 1/5
D. 1/2
E. 7/5

We are given that from a group of 3 doves and 2 rabbits, 2 animals will be randomly selected. We need to determine the probability that a matched pair will be pulled out of the hat.

In other words, we need to determine:

P(2 doves pulled) + P(2 rabbits pulled)

We can use combinations to determine the number of favorable outcomes (that 2 rabbits or 2 doves are selected) and the total number of outcomes (that 2 animals are selected from 5).

Let’s first determine the number of ways we can select 2 doves from 3:

# of ways to select 2 doves from 3 doves: 3C2 = 3

Next let’s determine the number of ways we can select 2 rabbits from 2:

# of ways to select 2 rabbits from 2 rabbits: 2C2 = 1

Now we can determine the number of ways to select 2 animals from a total of 5 animals:

5C2 = (5 x 4)/(2 x 1) = 10

Thus, the probability of selecting a matched pair is 3/10 + 1/10 = 4/10 = 2/5.

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A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
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xALIx 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?

A. 2/5
B. 3/5
C. 1/5
D. 1/2
E. 7/5

One approach is to apply probability rules

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)]
= 6/20 + 2/20
= 8/20
= 2/5

Originally posted by BrentGMATPrepNow on 05 Mar 2018, 15:05.
Last edited by BrentGMATPrepNow on 19 Sep 2021, 13:15, edited 1 time in total.
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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
TimeTraveller wrote:
3C2/5C2 + 2C2/5C2 = 3/10 + 1/10 = 4/10 = 2/5. Hence (A).

Hi

I am little confused in which case we select two things one by one and in which case both together. why are we not selecting 3c1*2C1 + 2C1*1C1 ?
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Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
You have 3 doves and 2 rabbits i.e. a total of 5 animals.
The probability that you pick a dove on your first pick is 3/5 (since there are 3 doves)
The probability that you pick a dove on your second pick too is 2/4 (because now only 2 doves are left after we picked a dove in the first pick. ALso only 4 animals are left to choose from)

Similarly,
The probability that you pick a rabbit on your first pick is 2/5 (since there are 2 rabbits)
The probability that you pick a rabbit on your second pick too is 1/4 (because now only 1 rabbit is left after we picked a rabbit in the first pick. Also only 4 animals are left to choose from)

Probability of picking a matched pair = 3/5 * 2/4 + 2/5 * 1/4 = 2/5

How come we don't have to account for the fact that you there are 3 doves and you only pick 2 of them?

i.e. For the doves I did this:

3/5 x 2/4 x 3! / 2! = 9/10

Ultimately my probability will end up being 1 which is wrong.
Re: A magician has five animals in his magic hat: 3 doves and 2 [#permalink]
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