Stats Question?

Question

According to the state lobster hatchery of Massachusetts, approximately 1 out of 20,000,000 lobster hatched in blue

A) how many lobster must be hatched before we can be at least 90% confident that at least 1 is blue
B) If 100,000,000 were hatched, what would be the probability that between 10 and 20 were blue?

Answer 1

The first answer is correct but rounded, 46051700.71.

For the second question, use the binomial distribution 9also called Bernoulli trial). Probability of success p=1/20,000,000, probability of failure q=1-p=19,999,999/20,000,000, chance of k successes in n trials is P(X=k)= nCk*p^k*q^(n-k). Then you add up P(X=10)+ P(X=11)+… +P(X=20). I had to program this into Maple because the numbers were so outrageous. I get a probability of (about, if I programmed correctly – high probability and if I typed all those silly 0’s and 9’s correctly-low probability) .03182797290.

The problem probably expected you to use the normal approximation to the binomial distribution, which uses mean mu=n*p=5 and deviation sigma=square root(n*p*q)= 2.236067922 approximately. Then P(10<=X<=20)=P(2.236068033<=z<= 6.708204099) using the key formula z=(x-mu)/sigma. This approximates to .0126736575.

Strangely, these two answers differ by a factor of about 2.5. Hmmm. I can’t find my error. Maybe it goes to show that the normal distribution should only be used in “reasonable” cases. I will review my work further and update if I find an error, but I don’t see one yet. If this is for a class, you might want to point this out to your instructor.

Answer 2

Probability of hatching a single blue lobster is 1/20,000,000, and of NOT catching the one is 19,999,999/20,000,000.

90% confident means that only 10% chance of NOT catching the blue one. How many tries do we need? - N trials defined as
(19,999,999/20,000,000)^N = 0.1

If you solve it using ln(), you will get an answer - 2.3*20,000,000 = 46,000,000 lobsters, which is quite a lot!

I am sure you can do the second one now..

Answer 3

You've posted 5 stats problems in a row about an hour ago. Clearly you have not tried to work on any of them but are just looking for someone to do your work.