1. ## Tree Problem

The Spruce tree drops seeds every fall with its leaves, 80% of the seeds survive the winter. For a seed that survives its first winter there is a 40% chance it sprouts and a 40% chance it remains dormant and a 20% chance it dies each winter. About 60% of the sprouts survive to become mature trees capable of producing their own seeds. This sprout to mature tree process takes 4 years. What is the critical number of seeds dropped per tree to ensure the survival of the spruce tree species?

2. Originally Posted by mathlete2
The Spruce tree drops seeds every fall with its leaves, 80% of the seeds survive the winter. For a seed that survives its first winter there is a 40% chance it sprouts and a 40% chance it remains dormant and a 20% chance it dies each winter. About 60% of the sprouts survive to become mature trees capable of producing their own seeds. This sprout to mature tree process takes 4 years. What is the critical number of seeds dropped per tree to ensure the survival of the spruce tree species?
The probability that a single dropped seed becomes a mature tree is (0.8)(0.4)(0.6) = 0.192.

Let n seeds drop and let X be the random variable number of seeds that become a mature tree.

Then X ~ Binomial(n, p = 0.192).

It now depends on how you define in, a probability sense, the statement ensure the survival of the spruce tree species .......

Some might argue that you now require E(X) = 1 ..... Others might argue that you require $\displaystyle \Pr(X \geq 1) > 0.99$, say ........

You'll have to decide for yourself (or get clarification from your instructor).