A forester finds that 4 out of every 9 pine trees in a certa

A forester finds that 4 out of every 9 pine trees in a certain region is infected with the pine beetle. If she randomly selects 9 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 exactly 2 trees will be infected with the pine beetle is: ???

B) The probability that at most 3 trees will be infected with the pine beetle is: ???

Solution

Here we use Binomial Distribution

B(x,n,p)= nC_x p^xq^n-x

n ---- number of trees = 9

p ---- probability = 4/9 = 0.4444

q = 1-p = 1-0.4444 = 0.5555

A) The probability that exactly 2 trees:

p( x = 2) = 9C₂(0.4444)²(0.5555)^9-2

             = 36(0.1975)(0.01632)

             = 0.1160

B) The probablity that at most 3 trees:

p(x <= 3) = p(x=0) + p(x=1) + p(x=2) = p(x=3)

              = 9C₀(0.4444)⁰(0.5555)^9-0 + 9C₁(0.4444)¹(0.5555)^9-1 + 9C₂(0.4444)²(0.5555)^9-2 + 9C₃(0.4444)³(0.5555)^9-3

              = 0.0050 + 0.0364 + 0.1160 + 0.2168

              = 0.3742