Answer
See explanation.
Work Step by Step
The error assumption was that the friend assumed that the events are mutually exclusive, so they could not happen together. If that were true $P(A\text{ or }B) = P(A) + P(B) = 0.3+0.7 = 1$. However, this isn't true. It's possible for there to be rain on both days, as well as for there to not be rain on either day.
The actual way to find $P(A\text{ or B})$ is $$P(A\text{ or B}) = P(A) + P(B) - P(A\text{ and }B).$$ Since $P(A\text{ and }B)$ is not $0$, $P(A\text{ or }$ will be less than $1$.
