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(1)? a. this value is the probability that one would encounter a contaminated cartridge in one hundred trials. b. this value is the probability that one would encounter a contaminated cartridge on the first trial. c. this value is the probability that one would encounter a non - contaminated cartridge on the first

Explanation:

To interpret \( p(1) \), we substitute \( x = 1 \) into the probability distribution formula \( p(x)=(0.24)(0.76)^{x - 1} \). When \( x = 1 \), the formula becomes \( p(1)=(0.24)(0.76)^{1 - 1}=(0.24)(0.76)^{0}=0.24\times1 = 0.24 \). The variable \( x \) represents the number of cartridges sampled until a contaminated one is found. So, \( x = 1 \) means we found a contaminated cartridge on the first trial. Option A is incorrect because it refers to one hundred trials, which is not related to \( x = 1 \). Option C is incorrect because \( p(1) \) is about finding a contaminated cartridge, not a non - contaminated one. Option B correctly interprets \( p(1) \) as the probability of encountering a contaminated cartridge on the first trial.

Answer:

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