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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.

But the answer provided is:

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
up vote 3 down vote accepted

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 $10$.

Your way considered the order, and was of course perfectly correct. It is hard to give failsafe guidelines for when your type of strategy is less efficient than the "choose" strategy. But rest assured: you know how to do the problem.

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@user6312: +1 Thank you very much for your time. That makes perfect sense now :) – Legend Apr 3 '11 at 2:15

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