probability of an event happening over a year
There are three options: either it happened in June (and didn't happen in July and August), in July alone or in August alone. The probability that it happened in one of the months, say June, and not happened in the others, is, similar to what you said: $0.10 \cdot 0.90 \cdot 0.90$ (there is a $90$% chance it won't happen). Therefore the solution is $3 \cdot 0.10 \cdot 0.90 \cdot 0.90$.