Given there are exactly 7 successes 3 minutes, what is the probability that they all came in the last minute? Suppose that phone calls arrive at a switchboard according to a Poisson Process at a rate of 2 per minute. Given there are exactly 7 calls in 3 minutes, what is the probability that they all came in the last minute? My idea is that this is similar to distinct balls in distinct buckets. So answer is 1/(3^7) = 0.0457%

That's correct. Conditioning a Poisson process on the number $n$ of events in a given time period yields $n$ events independently uniformly distributed over that time period, so each call independently has probability $\frac13$ of having arrived in the last minute.