Probability: 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?

Total number of ways in which any 3 random people can be chosen from 100 is $\binom{100}{3}$.

Now in our favourable cases, Michelle wins the prize, so in all favourable cases, Mitchelle is one of the 3 persons in the set which is chosen for winning. Now other 2 persons can be chosen in $\binom{99}{2}$ ways, so probability is

Yes you are right, but then we have to multiply denominator also by 3, and then 3 is canceled from numerator and denominator, resulting to same result 0.03.

But yes, we should multiply by 3 to consider all cases, although that doesn't change answer here.

Use binomial coefficients to compute the probability. Choose one of the prizes for Michelle and from the remaining 99 contestants choose the two other winners