A case of wine has 12 bottles, three of which contain spoiled wine. A sample of four bottles is randomly selected from the case.1. Find the probability distribution for X, the number of bottles of spoiled wine in the sample. 2. What are the mean and variance of X

(4Cx) × (0.25^x) × (0.75^4-x), mean =1, variance = 0.75

Step-by-step explanation:

Total bottle of wine = 12

Number of spoilt wine = 3

p(spoilt wine) = 3/12 = 1/4 = 0.25, q = 1 - 0.25 = 0.75.

n = number of bottles selected randomly = 4.

Since n = 4, then the experiment is of a binomial.

The probability mass function for a binomial distribution is given as

p(x=r) = nCr × p^r × q^n-r

The distribution for x is given below as

P(X=x) = (4Cx) × (0.25^x) × (0.75^4-x)

The mean for a binomial probability distribution is given as

Mean = np = 4×0.25 = 1

Variance = npq = 4×0.25×0.75 = 0.75

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