Question

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(x) = (0.24)(0.76)^{x - 1}, x = 1, 2, 3, ...
what is the correct interpretation for p(x ≥ 2)?

a. this value is the probability that one would first encounter a contaminated cartridge on at least one of the remaining trials after the first was attempted

b. this value is the probability that one would first encounter a contaminated cartridge on the first trial

c. this value is the probability that one would first encounter a contaminated cartridge on the second

Explanation:

• Analyze \( P(x\geq2) \): \( x \) is the number of cartridges sampled until a contaminated one is found. \( x\geq2 \) means the first contaminated cartridge is found at the 2nd, 3rd, 4th, etc., trial. In other words, the first trial ( \( x = 1 \)) did not yield a contaminated cartridge, and then we continue sampling (remaining trials) until we find one.
• Evaluate Option A: It says the probability of first encountering a contaminated cartridge on at least one of the remaining trials after the first attempt. This matches the interpretation of \( x\geq2 \) (since \( x = 1 \) is the first trial, \( x\geq2 \) is trials after the first until we find a contaminated one).
• Evaluate Option B: \( P(x = 1) \) is the probability of first encountering a contaminated cartridge on the first trial, not \( P(x\geq2) \), so B is wrong.
• Evaluate Option C: \( P(x = 2) \) is the probability of first encountering a contaminated cartridge on the second trial, but \( P(x\geq2) \) includes \( x = 2,3,4,... \), so C is wrong.

Answer:

A. This value is the probability that one would first encounter a contaminated cartridge on at least one of the remaining trials after the first was attempted