Question

It is known that 10% of the trees in a Timber Supply Area (TSA) are infested with the mountain pine beetle.
a) If 6 trees are randomly selected, what is the probability that 2 will be infested?

b) If 6 trees are randomly selected, what is the probability that 2 or more will be infested?

c) What is the mean and the standard deviation of the distribution described above (a and b)?

d) If trees are randomly selected one at a time, what is the probability that the 1st infested tree will be found when the 6th tree is checked?

e) If trees are randomly selected one at a time, what is the probability that the 4th healthy tree will be found when the 6th tree is checked?

f) A plot in this TSA contains 70 trees. What is the probability that exactly 5 trees are infested in any given plot?

g) In a given plot (70 trees), 10% of the trees are valued at $500, 700% are valued at $1,000, and 20% are valued at $2,000. What is the value of this plot? What is the standard deviation of the variable, value, for this plot?

Answer 1

Solution:-

a) If 6 trees are randomly selected, the probability that 2 will be infested is 0.0984 .

p = 0.10, n = 6

x = 2

By applying binomial distribution:-

P(x,n) = ⁿC_x*p^x*(1-p)^(n-x)

P(x = 2) = 0.0984

b)  If 6 trees are randomly selected, the probability that 2 or more will be infested is 0.1143 .

p = 0.10, n = 6

x = 2

By applying binomial distribution:-

P(x,n) = ⁿC_x*p^x*(1-p)^(n-x)

P(x > 2) = 0.1143

c) The mean and the standard deviation of the distribution described above is 0.60 and 0.7348 .

Mean = n*p

Mean = 6*0.10

Mean = 0.60

d) The probability that the 1st infested tree will be found when the 6th tree is checked is 0.05905

The first five trees will not be infested and 6th will be infested = (0.90)⁵*(0.10)

The probability that the 1st infested tree will be found when the 6th tree is checked is 0.05905