Problem Solving

BTGmoderatorLU wrote: ↑

Sat Dec 25, 2021 9:02 am

Source: GMAT Prep

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

The OA is A

We can 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

BTGmoderatorLU wrote: ↑

Sat Dec 25, 2021 9:02 am

Source: GMAT Prep

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

The OA is A

Another 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

Answer: A