The Question
Suppose that 100 people enter a contest and that different
winners are selected at random for first, second, and third
prizes. What is the probability that Michelle wins one of
these prizes if she is one of the contestants?
My Work
Our sample space is all the possible permutations of first second and third out of 100 contestants. Therefore $|S| = P(100,3)$
There are three possible scenarios where Michelle wins a prize. Therefore the probability that she wins a prize is $\frac{3}{P(100,3)}$
My Question
My book gave the answer $\frac{3}{100}$ why is the sample space only 100? This makes no sense to me. A sample size is all the possible outcomes of an experiment. The experiment was awarding first second and third too 100 contestants. $P(100,3)$ possible outcomes, right? Was my book wrong?
Best Answer
No, there are more than $3$ scenarios in which Michelle wins a prize. She can win first prize in $\frac{99\cdot 98}{2}$ since you have to include in you scenario the results of the other two picks, if you are going to divide by $\binom{100}{3}$. She can win second or third prize in the same number of ways.
So her chances of winning turn out to be $$\frac{\left(\frac{3\cdot 99 \cdot 98}{3!}\right)}{ \left(\frac{100\cdot 99 \cdot 98}{3!}\right)}=\frac{3}{100} $$