Seeds are often treated with a fungicide for protection in p

Seeds are often treated with a fungicide for protection in poor-draining, wet environments. In a small-scale trial prior to a large-scale experiment to determine what dilution of the fungicide to apply, six treated seeds and nine untreated seeds were planted in clay soil and the number of plants emerging from the treated and untreated seeds were recorded. Suppose the dilution was not effective and only four plants emerged. Let x represent the number of plants that emerged from treated seeds. (Round your answers to three decimal places.)

(a) Find the probability that x = 4.

b) Find

P(x 3).

(c) Find

P(2 x 3).

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( emerged plant) = 6/15 = 0.40
a)
P( X = 4 ) = ( 15 4 ) * ( 0.4^4) * ( 1 - 0.4 )^11
= 0.1268
b)
P( X < = 3) = P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 15 3 ) * 0.4^3 * ( 1- 0.4 ) ^12 + ( 15 2 ) * 0.4^2 * ( 1- 0.4 ) ^13 + ( 15 1 ) * 0.4^1 * ( 1- 0.4 ) ^14 + ( 15 0 ) * 0.4^0 * ( 1- 0.4 ) ^15   
= 0.0905
c)
P( X = 2 ) = ( 15 2 ) * ( 0.4^2) * ( 1 - 0.4 )^13
= 0.0219
P( X = 3 ) = ( 15 3 ) * ( 0.4^3) * ( 1 - 0.4 )^12
= 0.0634
P( 2 <=X<=3) = 0.0219+0.0634 = 0.0853