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 \\)

a. find \\( p(1) \\). interpret this result.
\\( p(1) = \square \\) (round to three decimal places as needed.)

Explanation:

Step1: Substitute x=1 into the formula

We have the probability distribution formula \( p(x) = (0.24)(0.76)^{x - 1} \). For \( x = 1 \), we substitute \( x = 1 \) into the formula: \( p(1)=(0.24)(0.76)^{1 - 1} \).

Step2: Simplify the exponent

Simplify the exponent \( 1 - 1 = 0 \). Any non - zero number to the power of 0 is 1, so \( (0.76)^{0}=1 \). Then \( p(1)=(0.24)\times1 = 0.240 \).

Answer:

0.240