# Confusion about calculating probability of at least one event occurring

The probability that Tom will win the Booker prize is 0.5, and the probability that John will win the Booker prize is 0.4. There is only one Booker prize to win. What is the probability that at least one of them wins the prize? (A) 0.2 (B) 0.4 (C) 0.7 (D) 0.8 (E) 0.9

My first approach: 1 - P (neither) = 1 - 0.5*0.6 = 0.7

Second approach: P(A) + P(B) - P(A and B) = 0.5 + 0.4 -0 = 0.9

Can someone please explain which one is correct and what is wrong with the other?

$P(A\cup B)^c = P(A^c\cap B^c) = P(A^c)P(B^c)$
Well, the first equality is just De Morgan's law, but the second one does not hold necessarily. This works if $A^c$ and $B^c$ are independent. In fact, it is definition of independent variables. You just showed that $A^c$ and $B^c$ aren't independent. See Wiki article.
EDIT: See also definition of probability measure here. In particular, the third axiom wants countable additivity of probability of disjoint events. Thus, $P(A\cup B) = P(A) + P(B)$ by that axiom since $A$ and $B$ are disjoint. More general statement for not necessarily disjoint sets that you use $P(A\cup B) = P(A) + P(B) - P(A\cap B)$ is actually a theorem in probability theory. Of course, you used it correctly, I am only telling you this so you can be more sure in your calculations in the future.