Contaminated gun cartridges. A weapons manufacturer uses a

Chapter 4, Problem 29E

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QUESTION:

Contaminated gun cartridges. A weapons manufacturer uses a liquid propellant to produce gun cartridges. During the manufacturing process, the propellant can get mixed with another liquid to produce a contaminated cartridge. A University of South Florida statistician, hired by the company to investigate the level of contamination in the stored cartridges, found that 23% of the cartridges in a particular lot were contaminated. Suppose you randomly sample (without replacement) gun cartridges from this lot until you find a contaminated one. Let x be the number of cartridges sampled until a contaminated one is found. It is known that the probability distribution for x is given by the formula

\(p(x)=(.23)(.77)^{x-1}, x=1,2,3, \ldots\)

a. Find p(1). Interpret this result.

b. Find p(5). Interpret this result.

c. Find \(P(x \geq 2)\). Interpret this result.

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QUESTION:

Contaminated gun cartridges. A weapons manufacturer uses a liquid propellant to produce gun cartridges. During the manufacturing process, the propellant can get mixed with another liquid to produce a contaminated cartridge. A University of South Florida statistician, hired by the company to investigate the level of contamination in the stored cartridges, found that 23% of the cartridges in a particular lot were contaminated. Suppose you randomly sample (without replacement) gun cartridges from this lot until you find a contaminated one. Let x be the number of cartridges sampled until a contaminated one is found. It is known that the probability distribution for x is given by the formula

\(p(x)=(.23)(.77)^{x-1}, x=1,2,3, \ldots\)

a. Find p(1). Interpret this result.

b. Find p(5). Interpret this result.

c. Find \(P(x \geq 2)\). Interpret this result.

ANSWER:

Step 1 of 3

We consider x as the number of cartridges sampled until a contaminated on is found.

We know that the formula for the probability distribution of x is

\(\mathrm{p}(\mathrm{x})=(0.23)(0.77)^{x-1}\) for x = 1, 2, 3,...

Our goal is :

a). We need to find \(P(x=1)\).

b). We need to find \(P(x=5)\).

c). We need to find \(P(x \geq 2)\).

a). Now we need to find p(1).

Put x = 1.

Then,

\(\begin{array}{l} \mathrm{p}(1)=(0.23)(0.77)^{1-1} \\ \mathrm{p}(1)=(0.23)(0.77)^{0} \\ \mathrm{p}(1)=(0.23)(1) \\ \mathrm{p}(1)=0.23 \end{array}\)

Therefore, p(1) = 0.23.

So the probability that one would encounter contaminated cartridges on the first trial is 0.23.

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