Question

Assume that eighteen samples are drawn from Lake Ontario. Each sample
of water has a 10% chance of containing a particular pollutant. Assume
that the samples are independent with regard to the presence of pollutant.
(a) What is the probability that exactly two do not contain the pollutant?
(b) What is the probability that at least four samples contain the pollutant?

Answer 1

X= containing a particular pollutant

n=18

p(success)=10%=0.10

q(failure)=1-p=1-0.10=0.90

X~binom(18,0.10)

P(X=x)=

(a) What is the probability that exactly two do not contain the pollutant?

P(exactly 2 do not contain pollutant)=1-p(exactly 2 contain pollutant)

P(X2)=1-P(X=2)

P(exactly 2 do not contain pollutant)=1-

P(exactly 2 do not contain pollutant)=1-

P(exactly 2 do not contain pollutant)=1-0.2835

P(exactly 2 do not contain pollutant)=0.7164

(b) What is the probability that at least four samples contain the pollutant?

P(X 4)=1-P(X<4)

P(X 4)=1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)]

P(X 4)=1-[+++]

P(X 4)=1-[+++]

P(X 4)=1-[0.1500+0.3001+0.2835+0.1680]

P(X 4)=1-0.9016

P(X 4)=0.0984

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