Question: Contaminated gun cartridges. A weapons manufacture uses a liquid propellant to produce gun cartridges. During the manufacturing process, the propellant can get mixed with another liquid to produce a contaminated cartridge. A University of South Florida statistician, hired by the company to investigate the level of contamination in the stored cartridges, found that \(23 \%\) of the cartridges in a 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:

\[p(x)=(.23)(.77)^{x-1}, \quad x=1,2,3, \ldots\]

(b) Find p(5)

(c) Find \(\Pr \left( X\ge 2 \right)\)

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