Problem Solving

lheiannie07 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)]
We get: 2/5 (or 0.4)

Answer: A


Cheers,
Brent

Hi lheiannie07,

We're told that a magician has 5 animals in his magic hat: 3 doves and 2 rabbits. We're asked - if he pulls two animals out of the hat at random, what is the chance that he will have a matched pair. The 'math' behind this question isn't too difficult, but you have to consider 2 outcomes to properly answer it.

To end up with a 'matching pair', the second animal chosen has to 'match' the first animal. The probability of a match occurring varies depending on whether the first animal is a dove or a rabbit. Keep in mind that once you pull out a dove or rabbit, there is one fewer animal remaining to choose for the second animal:

(1st dove)(2nd dove) = (3/5)(2/4) = 6/20

(1st rabbit)(2nd rabbit) = (2/5)(1/4) = 2/20

Total probability of pulling 2 matching animals = 6/20 + 2/20 = 8/20 = 2/5

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

lheiannie07 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

lheiannie07 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.

Alternate Solution:

The two events that satisfy the requirement of getting a matched pair are DD or RR.

The probability of DD is 3/5 x 2/4 = 6/20 =3/10.

The probability of RR is 2/5 x 1/4 = 2/20 = 1/10.

Since DD and RR are mutually exclusive events, the probability that either of these two events happens can be found simply by adding the individual probabilities, which is 3/10 + 1/10 = 4/10 = 2/5.

Answer: A