Problem : In a swimming race, the odds that A will win are $\frac23$ and the odds that B will win are $\frac14$. Find the odds that A or B win the race.
My Approach : I guess we can add up the odds of A and B to find the odds of winning of 'A and B'. That is, $\frac23+\frac14=\frac{11}{12}.$ Am I correct?
No you can't. There is a subtle distinction between odds and probability; $2:3$ odds is the probability $\frac{2}{2+3}$ and $1:4$ odds is $\frac{1}{1+4}$. With that in mind, you can sum the probabilities. The general formula for the probability A wins or B wins is $$P(A\cup B) = P(A) + P(B) - P(A \cap B)$$ where $P(A \cup B)$ denotes the probability A wins or B wins, and $P(A \cap B)$ is probability A wins and B wins, which is clearly impossible. Thus it reduces to $$= P(A) + P(B)$$ $$= \frac{2}{5} + \frac{1}{5}$$ $$= \frac{3}{5}$$