Whenever you need to find the probability of at least one thing happening, you can instead ask "What is the probability that none of them happen?" and subtract from $1$ (since the complementary event to "none happen" is "at least one happens").
For each of the individual events, we find the probability it does not happen by subtracting the probability that it does happen from 1. We have
- Event 1 doesn't happen: $19/21$
- Event 2 doesn't happen: $9/10$
- Event 3 doesn't happen: $8/15$
- Event 4 doesn't happen: $7/16$
- Event 5 doesn't happen: $7/10$
Since the events are independent, the probability no event happens is the product of the individual probabilities, which is $133/1000$.
Based on the calculation above $ \Pr(\text{at least one event}) = 1 - \Pr(\text{none of the events}) = 1 - \frac{133}{1000} = \frac{867}{1000} = 86.7\%. $