Beechwood forests in East Central Europe are being threatened by dynamic changes

Question

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 professors 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 20 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. What is the distribution of X? (Specify name and parameter values of the X distribution).
3. Use the PMF of X to calculate probability that in the sample there are trees damaged by fungi i.e. P(X 6= 0)
4. Given that there are trees damaged by fungi, what is the probability only one tree is damaged?

Explanation / Answer

1. For any randomly selected beech tree in this area, what is the chance it has been damaged by fungi?

p = 0.25 [ANSWER, GIVEN]

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2. Let X denote the number of beech trees, out of the sample, that are damaged by fungi. What is the distribution of X? (Specify name and parameter values of the X distribution).

It is a binomial distribution, with n = 20, p = 0.25. [ANSWER]

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3. Use the PMF of X to calculate probability that in the sample there are trees damaged by fungi i.e. P(X 6= 0)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    20      
p = the probability of a success =    0.25      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.003171212

Thus, P(at least one) = 1 - P(0) =   0.996828788 [ANSWER]

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4. Given that there are trees damaged by fungi, what is the probability only one tree is damaged?

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    20      
p = the probability of a success =    0.25      
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.021141413

Hence,

P(x=1|x>=0) = 0.021141413/0.996828788 = 0.02120867 [ANSWER]

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