Question

1. In a lake, Peter catches fishes in an average rate of 3 fishes per hour. (Assume Peter always catches one fish at a time and the events of catching each fish are independent.)
   (a) [10 points] At 2:00PM, if it’s known that Peter caught the previous fish at 1:50PM, what

   is the probability that he will catch the next fish before 2:30PM?
   (b) [10 points] Assume Peter starts catching fish at 9:00AM, what is the probability that he

   will catch the 2nd fish between 9:30 and 9:50AM?

Answer 1

a) It is known that Peter hasn't caught a fish from 1:50 PM to 2:00 PM.

The probability that Peter will catch the next fish before 2:30 PM is = The probability that Peter will catch at least one fish before 2:30 PM given that he hasn't caught a fish from 1:50 PM to 2:00 PM.

The probability of catching at least one fish from 1:50 PM to 2:30 PM

= 1 - The probability of catching no fish from 1:50 PM to 2:30 PM

= 1 -

3 / 60 minutes = 2/40 minutes =1.5/30 minutes = 1/20 minutes = 0.5/10 minutes

= 1 -

= 0.865

The probability of catching no fish from 1:50 PM to 2:00 PM is

The required probability that he will catch the next fish before 2.30 PM is

b) In order to catch the 2nd fish between 9:30 to 9:50AM, Peter should catch exactly 1 fish between 9 to 9:30AM.

The probability of catching exactly 1 fish between 9 to 9:30AM is

The probability of catching at least one fish between 9:30AM and 9:50AM is

= 1 - The probability of catching no fish between 9:30 AM to 9:50 AM

= 1 -

The probability that he will catch the 2nd fish between 9:30AM and 9:50AM is

= 0.335 * 0.632

= 0.212

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