Question 1201226: In a library box, there are 4 novels, 4 biographies, and 4 war history books. If Jack selects two books at random, what is the probability of selecting two different kinds of books in a row?
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
This is a good problem for solving by two of the basic methods of probability, to show that for certain types of problems one method is going to be a lot less work than the other.
First method: combinations
We need to choose (1 novel and 1 biography) or (1 novel and 1 war history) or (1 biography and 1 war history); and of course in each case 0 of the third type:
C(4,1)*C(4,1)*C(4,0)+C(4,1)*C(4,0)*C(4,1)+C(4,0)*C(4,1)*C(4,1) = (4)(4)(1)+(4)(1)(4)+(1)(4)(4) = 16+16+16 = 48.
So there are 48 ways to select two different kinds of books. The number of ways of choosing 2 of the 12 books is C(12,2) = 66, so the probability of choosing two different kinds of books is 48/66 = 8/11.
ANSWER: 8/11
Note that a different path to the solution still using combinations is to count the number of combinations with two different types of books as all 66 possible combinations, minus the ones with two books of the same type:
66 - (C(4,2)+C(4,2)+C(4,2)) = 66 - (6+6+6) = 66-16 = 48
And then again the probability of getting two different types of books is 48/66 = 8/11.
That is a very good way to solve the problem. However, another basic probability method solves the problem MUCH faster.
With the second basic probability method, we pick the books one at a time and consider the probability with each pick that we can still get the result we want.
We can choose any of the 12 books first; the probability that we can still get the desired final result is 12/12 = 1.
For the second book, there are 11 books remaining, of which 8 are a different type than the first. So the probability that we are okay on the second pick is 8/11.
And then the final answer for the problem is simply the product of the two probabilities: (1)(8/11) = 8/11.
If you are studying probability, you definitely want to know those two basic ways of solving many similar types of problems.
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