Question

3. Whole Foods has 18 two-liter bottles of a new Probiotic drink on the shelf, 4 of which had exceeded the shelf-life date.  You randomly select 3 bottles of the new drink without checking the expiration date.  Determine the probability that:

a.       a. None of the bottles have exceeded the expiration date.

b.      b. At least two bottles had exceeded the expiration date.

c.       c. Calculate the mean and standard deviation of this distribution.

Answer 1

Let us define a random variable 'x' for the number of expired bottles such that x ~ Binomial(n=3, p=4/18)

(a)

P(x=0) = ³C₀ .p⁰ .(1 - p)^{3 - 0} = 1 * (4/18)^0 * (1-4/18)^(3-0) = 0.471

(b)

P(x >= 2) = 1 - P(x < 2)

P(x < 2) = P(x=0) + P(x=1) = 0.471 + 3 * (4/18)^1 * (1-4/18)^(3-1) = 0.874

So,

P(x >= 2) = 1 - P(x < 2) = 1 - 0.874 = 0.126

(c)

Mean = n.p = 3*4/18 = 0.667

Variance = n.p.(1 - p) = 3*(4/18)*(1-4/18) = 0.5185

So, Stdev = √0.5185 = 0.72