Question

a) Within some forest, satellites show that there are approximately 345 different animals, 53 of which are a special type of bear. If the special bears are distributed throughout the forest uniformly, what is expected number of special bears that you expect to be in a random sample from the forest of size 30? Round your answer to the nearest integer.

b)Within some forest, satellites show that there are approximately 345 different animals, 53 of which are a special type of bear. If the special bears are distributed throughout the forest uniformly, what is the probability that a random sample of 30 contains at least the expected number of special bears (answer to previous problem) in the sample? Round your answer to 4 decimal places.

Answer 1

a)

Within some forest, out of 345 different animals, 53 are special types of bears.

They are distributed throughout the forest uniformly.

To find the expected number of special bears in a forest of size 30.

The probability of an animal being a special bear is 53/345=0.154.

So, we can say that if X be the random variable denoting the number of special bears in a forest of 30, X~Binomial(30,0.154).

So, expected number of special bears is

=30*0.154

=4.62

~5 approximately

(b)

To find the chance that a forest of size 30 contains atleast the expected number of special bears, ie. 5 bears.

ie.

P(X>=5)

=1-(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4))

=1-sumi((30Ci)(0.154)i(0.846)30-i)

=0.5001