Probability that the fisherman catches at least two fish

probability

Continued with question on Expected number of occurences in Poisson process that can stop under certain conditions. The question now is "Find the probability that he catches at least two fish."

From what I understand, this condition should happen after the first two hour, for he needs to catch at least one fish in the first hour. So, the second fish should be caught in the time afterwards.

However，the solution states:"If he catches at least two fish, he must have fished for exactly two hours…."

Why it is so? What have I missed?

Best Answer

... The fisherman always fishes for at least 2 hours. If during these 2 hours he catches at least 1 fish, he goes back home, else, he keeps fishing until he catches his first fish and then immediately leaves ...

He will always fish for at least two hours, regardless of how many fish he catches in that time. At the end of two hours:

  • If he has zero fish, he will continue fishing (after the two hours end) until he has one fish and then immediately stop and go home.
  • If he has one or more fish, he immediately stops (at the end of the two hours) and goes home.

In the first case, he never has two fish. So any scenario in which he has two or more fish must be the second case.

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