A forester finds that 4 out of every 15 pine trees in a cert

A forester finds that 4 out of every 15 pine trees in a certain region is infected with the pine beetle. If she randomly selects 8 pine trees
Note1: carry at least 6 digit precision for any intermediate calculations then round your answer to 4 decimal places
Note2: Any value that is less than 0.0001, (1 x 10^-4) , can be rounded to 0
A) The probability that at most 2 trees will be infected with the pine beetle is:  

B) The probability that exactly 1 trees will be infected with the pine beetle is:  

C) The probability that none of the trees will be infected with the pine beetle is:  

D) The probability that all 8 trees in the sample will be infected with the pine beetle is:

Solution

p = 4/15

n = 8

Number of trees that will be infected with the pine beetle follows Binomial Distribution with n= 8 and p = 4/15

A. P(X <= 2) = P(X=0) + P(X=1) + P(X=2) = (1-p)⁸ + 8C1*p*(1-p)⁷ + 8C2*p²*(1-p)⁶ = 0.6366

B. P(X=1) = 8C1*p*(1-p)⁷ = 0.2433

C. P(X=0) = (1-p)⁸ = 0.0836

D. P(X=8) = p⁸ = 0.00002 = 0