Questions: You can retry this question below Assume that 12 jurors are randomly selected from a population in which 77% of the people are Mexican-Americans. Refer to the probability distribution table below and find the indicated probabilities. Note that 0+ means that the probability rounds off to zero at 4 decimal places. For the purposes of your calculations, you may use 0+=0. x P(x) 0 0+ 1 0+ 2 0+ 3 0.0002 4 0.0014 5 0.0073 6 0.0285 7 0.0818 8 0.1712 9 0.2547 10 0.2558 11 0.1557 12 0.0434 Find the probability of exactly 5 Mexican-Americans among 12 jurors. P(x=5)= Find the probability of 5 or fewer Mexican-Americans among 12 jurors.

You can retry this question below

Assume that 12 jurors are randomly selected from a population in which 77% of the people are Mexican-Americans. Refer to the probability distribution table below and find the indicated probabilities. Note that 0+ means that the probability rounds off to zero at 4 decimal places. For the purposes of your calculations, you may use 0+=0.

x P(x)
0 0+
1 0+
2 0+
3 0.0002
4 0.0014
5 0.0073
6 0.0285
7 0.0818
8 0.1712
9 0.2547
10 0.2558
11 0.1557
12 0.0434

Find the probability of exactly 5 Mexican-Americans among 12 jurors.
P(x=5)=

Find the probability of 5 or fewer Mexican-Americans among 12 jurors.

Transcript text: You can retry this question below Assume that 12 jurors are randomly selected from a population in which $77 \%$ of the people are Mexican-Americans. Refer to the probability distribution table below and find the indicated probabilities. Note that $0+$ means that the probability rounds off to zero at 4 decimal places. For the purposes of your calculations, you may use $0+=0$. \begin{tabular}{|r|r|} \hline$x$ & $P(x)$ \\ \hline 0 & $0+$ \\ \hline 1 & $0+$ \\ \hline 2 & $0+$ \\ \hline 3 & 0.0002 \\ \hline 4 & 0.0014 \\ \hline 5 & 0.0073 \\ \hline 6 & 0.0285 \\ \hline 7 & 0.0818 \\ \hline 8 & 0.1712 \\ \hline 9 & 0.2547 \\ \hline 10 & 0.2558 \\ \hline 11 & 0.1557 \\ \hline 12 & 0.0434 \\ \hline \end{tabular} Find the probability of exactly 5 Mexican-Americans among 12 jurors. \[ P(x=5)= \] $\square$ Find the probability of 5 or fewer Mexican-Americans among 12 jurors.

Solution

Solution Steps

Step 1: Probability of Exactly 5 Mexican-Americans

To find the probability of exactly 5 Mexican-Americans among 12 jurors, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

$ n = 12 $ (the number of trials),
$ x = 5 $ (the number of successes),
$ p = 0.77 $ (the probability of success),
$ q = 1 - p = 0.23 $ (the probability of failure).

Calculating this gives:

\[ P(X = 5) = 0.0073 \]

Step 2: Probability of 5 or Fewer Mexican-Americans

To find the probability of having 5 or fewer Mexican-Americans, we sum the probabilities from $ P(X = 0) $ to $ P(X = 5) $:

\[ P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) \]

Calculating these probabilities, we find:

$ P(X = 0) = 0.0 $
$ P(X = 1) = 0.0 $
$ P(X = 2) = 0.0 $
$ P(X = 3) = 0.0002 $
$ P(X = 4) = 0.0014 $
$ P(X = 5) = 0.0073 $

Summing these values:

\[ P(X \leq 5) = 0.0 + 0.0 + 0.0 + 0.0002 + 0.0014 + 0.0073 = 0.0089 \]

Final Answer

The results are:

The probability of exactly 5 Mexican-Americans among 12 jurors is $ P(x=5) = 0.0073 $.
The probability of 5 or fewer Mexican-Americans among 12 jurors is $ P(x \leq 5) = 0.0089 $.

Thus, the final answers are: \[ \boxed{P(x=5) = 0.0073} \] \[ \boxed{P(x \leq 5) = 0.0089} \]

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