In: Statistics and Probability
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? Please show step by step process for
arriving at such answer
(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? Please show step by step process for arriving at such
answer
Answer:
Given Data
a) 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.
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
= 2/40 minutes
= 1.5/30 minutes
= 1/20 minutes
= 0.5/10 minutes
= 0.865
The probability of catching no fish from 1:50 PM to 2:00 PM is
= 0.606
The required probability that he will catch the next fish before 2.30 PM is
= 0.7012
b) 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
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
= 0.335
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
= 0.632
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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