Question

A tap water quality survey indicates that 45% of the houses in a city have a level of lead in their tap water higher than the maximum suggested by the Authority. For a random sample of 6 houses, find the probability that a. All the 6 houses have lead levels above the maximum. b. Exactly 3 houses have a lead level above the maximum. c. At most 1 house has a lead level above the maximum.

Answer 1

it is the case of binomial distribution:

X=houses in a city have a level of lead in their tap water higher than the maximum

p= sucess:houses in a city have a level of lead in their tap water higher than the maximum =45%=0.45

q=failure:houses in a city do not have a level of lead in their tap water higher than the maximum=1-p=1-0.45=0.55

n=6

X~binom(6,0.45)

we know:

(a) All the 6 houses have lead levels above the maximum:

P(X=6)=₆C₆*(0.45)⁶(0.55)⁰

P(X=6)=1*0.008303*1

P(X=6)=0.008303

(b) Exactly 3 houses have a lead level above the maximum:

P(X=3)=₆C₃*(0.45)³(0.55)³

P(X=3)=20*(0.09112)*(0.1663)

P(X=3)=0.303065

(c) At most 1 house has a lead level above the maximum.

P(X1)=P(X=0)+P(X=1)

=₆C₀*(0.45)⁰*(0.55)⁶+₆C₁*(0.45)¹*(0.55)⁵

=1*1*(0.02768)+6*(0.45)*(0.05032)

=0.02768+0.135864

P(X1) =0.163544

please rate my answer and comment for doubts.