91Ó°ÊÓ

The probability of finding 1 contaminated gun cartridge is calculated below:

$$\begin{gathered} \mathbf{\text{p}}\mathbf{\left(}\mathbf{\text{x}}\mathbf{\right)}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{x}}\mathbf{−}\mathbf{\text{1}}} \\ \mathbf{\text{p}}\mathbf{\left(}\mathbf{\text{1}}\mathbf{\right)}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{−}\mathbf{\text{1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{0}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.23}} \end{gathered}$$

Here the value $\mathbf{\text{p}}\mathbf{\left(}\mathbf{\text{1}}\mathbf{\right)}$from the calculation is 0.23.This value indicates that the probability of finding a contaminated cartridge from the sample of cartridges is 0.23.

The probability of finding 5 contaminated cartridges from the sample is calculated below:

$$\begin{gathered} \mathbf{\text{p}}\mathbf{\left(}\mathbf{\text{x}}\mathbf{\right)}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{x-1}}} \\ \mathbf{\text{p}}\mathbf{\left(}\mathbf{\text{5}}\mathbf{\right)}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{5-1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{4}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.08}} \end{gathered}$$

Here the value of $\mathbf{p}\mathbf{\left(}\mathbf{5}\mathbf{\right)}$ the above calculation is found to be 0.08.This value indicates that the probability of finding five contaminated gun cartridges from the sample of several cartridges is 0.08.

The probability of finding a maximum of two contaminated cartridges is calculated below:

$$\begin{gathered} \mathbf{\text{p}}\mathbf{(}\mathbf{\text{x}}\mathbf{â‰\yen }\mathbf{\text{2}}\mathbf{)}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{x-1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{p}}\mathbf{\left(}\mathbf{\text{1}}\mathbf{\right)}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{1-1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{0}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.23}} \end{gathered}$$

$$\begin{gathered} \mathbf{\text{p}}\mathbf{\left(}\mathbf{\text{2}}\mathbf{\right)}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{2-1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\mathbf{\left(}\mathbf{\text{.23}}\mathbf{\right)}{\mathbf{\left(}\mathbf{\text{.77}}\mathbf{\right)}}^{\mathbf{\text{1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.18}} \\ \mathbf{\text{p}}\mathbf{(}\mathbf{\text{x}}\mathbf{â‰\yen }\mathbf{\text{2}}\mathbf{)}\mathbf{\text{=p}}\mathbf{\left(}\mathbf{\text{1}}\mathbf{\right)}\mathbf{\text{+p}}\mathbf{\left(}\mathbf{\text{2}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.23+0.18}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.41}} \end{gathered}$$

Here, the calculation$\mathbf{\text{p}}\mathbf{(}\mathbf{\text{x}}\mathbf{â‰\yen }\mathbf{\text{2}}\mathbf{)}$is found to be 0.41.This value indicates that the probability of finding a maximum of two contaminated gun cartridges by the researchers from the sample is 0.41.