Credit for this problem goes to Graham White. Thanks!

Balls in an urn

An urn contains 20 balls, of which 17 are red and 3 are green. You remove one at a time, uniformly at random, until you remove a ball of a different color than the first ball you removed. At that point, you replace that last ball (the one that was a different color), and start over with the remaining balls.

For example, you might remove red, red, green, then replace green, then remove red, red, red, green, replace green, remove green, green, red, replace red, etc.

What is the chance that the last ball removed is green?