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In a Bernoulli trial, the likelihood of the number of successive failures before a success is obtained is represented by a geometric distribution, which is a sort of discrete probability distribution. A Bernoulli trial is a test that can only have one of two outcomes: success or failure.

Variable X has a geometric distribution with the probability of success $0.173$.

Therefore, the graphic presentation is as follow:

The probability that we need to test 30 people to find one with HIV is calculated as:

$$\begin{gathered} P(X=30)={(1-0.173)}^{x-1}脳0.173 \\ P(X=30)={(0.827)}^{29}(0.173) \\ P(X=30)=0.0007 \end{gathered}$$

The probability that we must ask ten people is as follow:

$$\begin{gathered} P(X=10)={(1-0.173)}^{n-1}(0.173) \\ P(X=10)={(0.827)}^9(0.173) \\ P(X=10)=0.0313 \end{gathered}$$

The mean of the variable with geometric distribution with parameter $p=0.173$

Mean$=\frac{1}{p}$

$$\begin{gathered} Mean=\frac{1}{0.173} \\ Mean=5.7803 \end{gathered}$$

Standard deviation is $未=\sqrt{\frac{1-p}{p^2}}$where$p=0.173$

$$\begin{gathered} 未=\sqrt{\frac{1-p}{p^2}} \\ 未=\sqrt{\frac{1-0.173}{{(0.173)}^2}} \\ 未=5.2566 \end{gathered}$$