Boat sighting chance over ten minutes

Question

Your boat has broken down at sea. Passing boats appear as a homogeneous Poisson process, and you know there is a $73.7856\%$ chance of seeing at least one boat in any one-hour window. What is the probability of seeing at least one boat in the next $10$ minutes?

Short answer

$0.2$ . For a Poisson process, the chance of no boat over an hour is the product of the chances over six disjoint 10-minute windows, so $p^6=1-0.737856=0.262144$ , giving $p=0.8$ . The probability of at least one boat in 10 minutes is $1-0.8=0.2$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Defining the Poisson Process: Three Equivalent Constructions
Independence
The Exponential Distribution and Memorylessness

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