# Quick probability question on matching

I am solving some practice questions while revising probabilities and wanted to confirm my methodology. The question is as follows:

A magician has five animals: 2 doves and 3 rabbits- If he pulls 2 animals out at random, what is the chance that he will get a matched pair.

The way I solved it:

Probability of matched pair= Probability of 2 doves (2/5*1/4= 2/20)+ Probability of 2 rabbits (3/5*2/4) = 2/20+ 6/20= 8/20.

Number of ways/ total number of ways= 4/10.

Conceptually both seem appropriate but the answers are different: Is it just that the answer was written as 4/10 instead of 8/20 or can someone please tell me if I am missing something?

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Why are you saying the answers are different? Both are the same, 2/5, and both are appropriate ways of solving this problem. – Alon Amit Apr 3 '11 at 2:05
@Alon Amit: Actually you are right. The answer is 2/5 but what I am interested in knowing is if my methodology is accurate or is there some other way the answer could have been derived? – Legend Apr 3 '11 at 2:07
As shown in user6312's post your methodology is accurate and there is another way the answer could have been derived. They are certainly not exclusive events. +1 for showing some thought. – Ross Millikan Apr 3 '11 at 3:52

The answers are the same, and both correct. For an explanation of the way the textbook did it, note that there are $\binom{5}{2}$ ways of choosing $2$ objects from $5$. These $10$ ways are all equally likely.
How many ways to get a match? Double Dove ($1$ way) or Double Rabbit. There are $\binom{3}{2}$ (namely $3$) ways of choosing $2$ rabbits from $3$. So the total number of "good" choices is $1+3$. Now divide by $10$.