Swans

A flock of n swans is sitting peacefully on a pond. Then, suddenly, one of the swans hears something in a bush at the water’s edge, and it takes off and flies away.

Suppose that if k swans have flown away, the probability of another swan subsequently joining them is 1-\frac{1}{k+3} (so the more swans fly, the more likely it is that more swans fly also). What is the probability that the entire flock of n swans flies away once the first one takes off?

The solution can be found here. Happy puzzling!

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