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A large lake is stocked with 10% catfish. You go out on the lake to fish, and each fish you catch that is not a catfish is thrown back into the lake. (Assume that each time you catch a fish, it is independent of the other fish you have caught.) Let X be the number of tries it takes until you catch your first catfish.

(a) What is the average (expected) value for X?
(b) What is the probability that X is at most 3?

Answer :

Answer:

a.) Expected value = 10.

b.) The probability that X is at most 3  = 0.271

Step-by-step explanation:

A large lake is stocked with 10% catfish.

Every fish caught that is not a catfish is thrown back.

X is the number of tries before we catch our first catfish.

We can say that the distribution is X [tex]\sim[/tex]  Geo( p = 0.1).

In a geometric distribution we can have only two outcomes of a particular trial and that is it will either be a success or a failure.

a.) The average or expected value is  [tex]E(X) = \frac{1}{p} = \frac{1}{0.1} = 10[/tex].

b.) X is at most three

    p(X ≤ 3)    = p(X = 1) + p(X = 2) + p(X = 3)

                     =  0.1      + ( 0.1 × 0.9)  + ( 0.1 × [tex](0.9)^2[/tex])

                     =   0.1 +  0.09  + 0.081

                     = 0.271

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