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# A weapons manufacturer 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$p ( x ) = ( .23 ) ( .77 ) ^ { x - 1 } , x = 1,2,3 , \ldots$Find$P ( x \geq 2 )$Interpret this result.

Solution

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Given:

\begin{aligned} p(x)&=0.23(0.77)^{x-1} \\ x&=1,2,3,... \end{aligned}

We need to determine the probability $P(X\geq 2)$.

To derive this probability, we will use the complement rule $P(A^C)=P(\text{ not A})=1-P(A)$.

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