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
