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.
$p(x) = (0.24)(0.76)^{x - 1}, x = 1, 2, 3, \dots$
this value is the probability that one would encounter a contaminated cartridge on the first trial
this value is the probability that one would encounter a non - contaminated cartridge on the first trial
b. find $p(4)$. interpret this result.
$p(4)=\square$ (round to three decimal places as needed.)

Explanation:

Step1: Identify the formula and x value

We are given the probability distribution formula \( p(x) = (0.24)(0.76)^{x - 1} \) and we need to find \( p(4) \), so we substitute \( x = 4 \) into the formula.

Step2: Substitute x = 4 into the formula

Substitute \( x = 4 \) into \( p(x) \):
\( p(4) = (0.24)(0.76)^{4 - 1} \)
First, calculate the exponent: \( 4 - 1 = 3 \)
Then, calculate \( (0.76)^3 \): \( 0.76^3 = 0.76\times0.76\times0.76 = 0.438976 \)
Now, multiply by 0.24: \( 0.24\times0.438976 = 0.10535424 \)

Step3: Round to three decimal places

Rounding \( 0.10535424 \) to three decimal places gives \( 0.105 \)

Answer:

\( p(4) = 0.105 \)