Confusion about calculating probability of at least one event occurring?

Question

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?

Answer 1

Your second approach is correct. That is

P(A ⋃ B)

= P(A) + P(B) - P(A ⋂ B)

= 0.5 + 0.4 - 0

= 0.9

In the first approach you are assuming A^c (Tom will NOT win the prize) and B^c (John will NOT win the prize) to be independent of each other but this assumption is false i.e. P(A^c ⋂ B^c) ≠ P(A^c) * P(B^c)

1 - P(Neither of them will win the prize)

= 1 - P(A^c ⋂ B^c)

= 1 - P((A ⋃ B)^c)

= 1 - [1 - P(A ⋃ B)]

= P(A ⋃ B)

Answer 2

The latter is correct, since this question states Tom and John winning are mutually exclusive events. Since there is only 1 prize to win, the probability of someone else winning the prize is 1 - 0.5 - 0.4 = 0.1