I think the easiest way to go about this is to figure out what the probability is of none of them making it... Since these are independent events, this scenario is much like flipping coins. For example, let's have two coins and study the probability that at least one of the coins turns up heads. There are four scenarios in this case:

TT
TH
HT
HH

It is easy to see that the probability of at least one of the coins turning up heads is 3/4. Conversely, if you figure out that the probability of both coins turning out tails is 1/4, you just need to subtract that probability from 1.

Now turning to your example, the probability of none of the pitchers getting to 300 wins is 59,049/100,000 or approximately 59% (9/10 to the 5th power).

This means that there is approximately a 41% chance that at least one of the five will win 300 games.