Problem 7. Beechwood forests in East Central Europe are being threatened by dyna
ID: 3329230 • Letter: P
Question
Problem 7. Beechwood forests in East Central Europe are being threatened by dynamic changes in land ownership and economic upheaval. The current status of the beech tree species in this area was evaluated by Hungarian university pro-fessors in Applied Ecology and Environmental Research (Vol 1, 2003). The researchers found that 25% of the beech trees in East Central Europe have been damaged by fungi. Consider a sample of 8 beech trees from this area. 1. For any randomly selected beech tree in this area, what is the chance it has been damaged by fungi? 2. Let X denote the number of beech trees, out of the sample, that are damaged by fungi. Write down the PMF of X). 3. Use the PMF of X to calculate probability that in the sample there are trees damaged by fungi i.e. P(X >0) 4. Given that there are trees damaged by fungi, what is the probability only one tree is damaged?Explanation / Answer
1) P(damaged by fungi) = 0.25
2) This is a binomial distribution.
p = 0.25
n = 8
P(X = x) = 8Cx * 0.25x * (1-0.25)8-x = 8Cx * 0.25x * 0.758-x
3) P(X > 0) = 1 - P(X = 0)
= 1 - 8C0 * 0.250 * 0.758
= 1 - 0.1001
= 0.8999
4) P(X = 1 | X > 0) = P(X = 1 and X > 0) / P(X > 0)
= P(X = 1) / P(X > 0)
= 8C1 * 0.251 * 0.757 / 0.8999
= 0.2670 / 0.8999
= 0.2967