Question in the book:
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:
Being first: 1/100
Being second: 1/99
Being third: 1/98
So total probablity is 1/100 + 1/99 + 1/98
Book Answer:
But the book says 3/100, I understand why kind of, but I don't understand what I did wrong in my work?
Best Answer
The probability of being second/third is not $\frac 1{99}, \frac 1{98}$. Then, is the probability of coming $99$th $\frac{1}{100-99} = 1$? Clearly not,right?
Choosing a position for Michelle does not depend upon what position it is. The probability that she comes first, is equal to the probability that she comes second, is equal to the probability that she comes third, because in each case, we have to choose one place out of hundred : whether that is the first place, or second place, or third place, or seventy-seventh place, it is still one place out of hundred. Therefore, the probability of each of these events is $\frac 1{100}$, and given that coming first,second and third are mutually exclusive events gives the answer $\frac 3{100}$.