Question

In: Statistics and Probability

A weapons manufacturer uses a liquid propellant that can get mixed with another liquid to produce...

A weapons manufacturer uses a liquid propellant that can get mixed with another liquid to produce a contaminated cartridge. A statistician found that 24​% of the cartridges in the 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 shown below. Complete parts a through c. p left parenthesis x right parenthesis equals left parenthesis 0.24 right parenthesis left parenthesis 0.76 right parenthesis Superscript x minus 1​, xequals​1, 2​ ,3 ,... a. Find ​p(1​). Interpret this result. ​p(1​)equals nothing ​(Round to three decimal places as​ needed.) What is the correct interpretation for ​p(1​)? A. This value is the probability that one would encounter a contaminated cartridge on the first trial. B. This value is the probability that one would encounter a contaminated cartridge in one hundred trials. C. This value is the probability that one would encounter a​ non-contaminated cartridge on the first trial. b. Find ​p(5​). Interpret this result. ​p(5​)equals nothing ​(Round to three decimal places as​ needed.) What is the correct interpretation for ​p(5​)? A. This value is the probability that one would first encounter 5 contaminated cartridge in one hundred trials. B. This value is the probability that one would first encounter a contaminated cartridge on the fifth trial. C. This value is the probability that one would first encounter a​ non-contaminated cartridge on the fifth trial. c. Find ​P(xgreater than or equals​2). Interpret this result. Upper P left parenthesis x greater than or equals 2 right parenthesisequals nothing ​(Round to three decimal places as​ needed.) What is the correct interpretation for ​P(xgreater than or equals​2)?

Solutions

Expert Solution

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 shown below.

a. Find ​p(1​).

We get ​p(1​)= 0.240

According to the event X, defined in the problem,

A. This value is the probability that one would encounter a contaminated cartridge on the first trial.

b.

We get ​p(5​)= 0.080

B. This value is the probability that one would first encounter a contaminated cartridge on the fifth trial.

c. Find

= 1 - 0.24

= 0.76

= 0.760

Interpretation:

C. This value is the probability that one would first encounter a contaminated cartridge on 2nd or later trial, which is equal to the complementary of the probability that one would encounter a contaminated cartridge on the 1st trial itself.

orchestra answered 1 year ago
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