Mathematics, 31.03.2020 00:01 noname61

A cheese can be classified as either raw dash milk or pasteurized. Suppose that 99% of cheeses are classified as pasteurized. (a) Two cheeses are chosen at random. What is the probability that both cheeses are pasteurized? (b) Eight cheeses are chosen at random. What is the probability that all eight cheeses are pasteurized? (c) What is the probability that at least one of eight randomly selected cheeses is raw dash milk? Would it be unusual that at least one of eight randomly selected cheeses is raw dash milk?

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Answer from: help977

(a) The probability that both cheeses are pasteurized is 0.9801.

(b) The probability that all eight cheeses are pasteurized is 0.9227.

(c) The probability that at least one of eight randomly selected cheeses is raw dash milk is 0.0773. The event is not unusual.

Step-by-step explanation:

Let X = number of cheese that are pasteurized.

The probability that a cheese is pasteurized is, p = 0.99.

A cheese being pasteurized is independent of the others.

The random variable X follows a Binomial distribution with parameters n and p.

The probability mass function of X is:

$P(X=x)={n\choose x}0.99^{x}(1-0.99)^{n-x};\ x=0,1,2,3...$

(a)

Compute the probability that both cheeses are pasteurized from the random selected sample of 2 cheese as follows:

$P(X=2)={2\choose 2}0.99^{2}(1-0.99)^{2-2}=1\times 0.9801\times 1=0.9801$

Thus, the probability that both cheeses are pasteurized is 0.9801.

(b)

A random sample of 8 cheeses are selected.

Compute the probability that all eight cheeses are pasteurized as follows:

$P(X=8)={8\choose 8}0.99^{8}(1-0.99)^{8-8}\\=1\times 0.922745\times 1\\=0.922754\\\approx0.9227$

Thus, the probability that all eight cheeses are pasteurized is 0.9227.

(c)

Let X₁ = number of cheese that are raw dash milk.

The probability that a cheese is raw dash milk is, q = 1 - p = 1 - 0.99 = 0.01.

The random variable X₁ also follows a Binomial distribution with parameters n and q.

Compute the probability that at least one of eight randomly selected cheeses is raw dash milk as follows:

P (X₁ ≥ 1) = 1 - P (X₁ < 1)

= 1 - P (X₁ = 0)

$=1-{8\choose 0}0.01^{0}(1-0.01)^{8-0}\\=1-(1\times 1\times 0.9227)\\=1-0.9227\\=0.0773$

Thus, the probability that at least one of eight randomly selected cheeses is raw dash milk is 0.0773.

An unusual event has a very low probability of success, mostly less than 0.05.

The probability that at least one of eight randomly selected cheeses is raw dash milk is 0.0773.

This probability is more than 0.05.

Thus, the it would not be unusual hat at least one of eight randomly selected cheeses is raw dash milk.

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