Question

11. A pharmacist receives a shipment of 22 bottles of a drug and has 3 of the bottles tested. If 5 of the 22 bottles are contaminated, what is the probability that at least 2 of the tested bottles are contaminated? Express your answer as a fraction or a decimal number rounded to four decimal places.

Answer 1

There are total 22 bottles of which 5 bottles are contaminated.

That is (22-5) = 17 bottles are pure (Not contaminated)

Let x be the number of bottles are contaminated.

We have to calculated P( X >= 2) = ?

Number of ways to select 2 of the 3 bottles are contaminated = ⁵C₂ * ¹⁷C₁

Similarly,

Number of ways to select 3 of the 3 bottles are contaminated = ⁵C₃ * ¹⁷C₀

Total number of select 3 bottles from 22 bottles = ²²C₃

P( X >= 2) = P( X = 2) + P( X = 3)

Using hyper geometric distribution,

P( X >=2) = [ ⁵C₂ * ¹⁷C₁ + ⁵C₃ * ¹⁷C₀ ] / ²²C₃

= 180 / 1540

= 0.1169