At Fontaine Lake Camp on Lake Athabasca in northern Canada h

At Fontaine Lake Camp on Lake Athabasca in northern Canada, history shows that about 37% of the guests catch lake trout over 20 pounds on a 4-day fishing trip. Let n be a random variable that represents the first trip to Fontaine Lake Camp on which a guest catches a lake trout over 20 pounds.

(a) Write out a formula for the probability distribution of the random variable n.
P(n) =  

(b) Find the probability that a guest catches a lake trout weighing at least 20 pounds for the first time on trip number 3. (Round your answer to three decimal places.)


(c) Find the probability that it takes more than three trips for a guest to catch a lake trout weighing at least 20 pounds. (Round your answer to three decimal places.)


(d) What is the expected number of fishing trips that must be taken to catch the first lake trout over 20 pounds? Hint: Use for the geometric distribution and round. (Round your answer to two decimal places.)
trips

Solution

Sol)

a) The random variable x satisfies the requirements for a Geometric distribution since we are looking at the “first time” an event happens.

The pmf of Geometric distribution is p(x) =p q^k-1 where p=0.37 and q=1-p=0.73

b) probability that a guest catches a lake trout weighing at least 20 pounds for the first time on trip number 3

The probability of “success” on one trip is p = 0.37, and q = 1p = 0.73. The probability that a guest catches a lake trout weighing at least 20 pounds for the first time on trip number 3 is P(3) = 0.37 · 0.73 ³¹ = 0.197

C)

The probability that it takes more than three trips for a guest to catch a lake trout weighing at least 20 pounds is P(x > 3) = 1 P(x 3) = 1 P(1) P(2) P(3) = 1 0.37 * 0.73 ¹¹ 0.37 * 0.73 ²¹ 0.37 * 0.73 ³¹

  P(x > 3) = 0.163

D) expected number of fishing trips that must be taken to catch the first lake trout over 20 pounds

Mean of Exponential distribution is 1/p =1/037 =2.703