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 $p(4)$?
\\(\bigcirc\\) a. this value is the probability that one would first encounter a non - contaminated cartridge on the fourth trial.
\\(\bigcirc\\) b. this value is the probability that one would first encounter a contaminated cartridge on the fourth trial.
\\(\bigcirc\\) c. this value is the probability that one would first encounter 4 contaminated cartridge in one hundred

Explanation:

The probability distribution \( p(x) = (0.24)(0.76)^{x - 1} \) models the number of trials \( x \) until the first contaminated cartridge is found (geometric distribution). For \( p(4) \), \( x = 4 \) means we are finding the probability that the first contaminated cartridge is encountered on the 4th trial. Option A is incorrect as it refers to a non - contaminated cartridge, and Option C is incorrect as it misinterprets the number of contaminated cartridges and the context.

Answer:

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