Eighty percent of households say they would feel secure if they had 50,000 in sa

iohanetc 2021-10-01 Answered

Eighty percent of households say they would feel secure if they had 50,000 in savings. You randomly select 8 households and ask them if they would feel secure if they had $50,000 in savings. Find the probability that the number that say they would feel secure is exactly five, (b) more than five, and (c) at most five
c.) Find the probability that the number that say they would feel secure is at most five. \(\displaystyle{P}{\left({<}{5}\right)}=\)
(round to three decimal places as needed).

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Expert Answer

wheezym
Answered 2021-10-02 Author has 13497 answers

Step 1
Given data
Probability of households feeling insecure \(\displaystyle{p}={0.80}\)
\(\displaystyle{n}={8}\)
a) probability that the number that say they would feel secure is exactly five is given by
Applying Binomial theorem
Probability of x success out of n trial is given by
\(P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right) \times p^{x} \times (1-p)^{n-x}\)
\(\Rightarrow P(X=5)=\left(\begin{array}{c}8\\ 5\end{array}\right) \times 0.8^{5} \times (1-0.8)^{8-5}\)
\(\displaystyle={\frac{{{8}!}}{{{5}!\times{\left({8}-{5}\right)}!}}}\times{0.8}^{{{5}}}\times{\left({1}-{0.8}\right)}^{{{8}-{5}}}\)
\(\displaystyle={0.1468}\)
Step 2
b) probability that the number that say they would feel secure is more than five is given by
Applying Binomial theorem
Probability of x success out of n trial is given by
\(P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right) \times p^{x} \times (1-p)^{n-x}\)
\(\displaystyle\Rightarrow{P}{\left({X}{>}{5}\right)}={P}{\left({X}={6}\right)}+{P}{\left({X}={7}\right)}+{P}{\left({X}={8}\right)}\)
\(\Rightarrow \left(\begin{array}{c}8\\ 6\end{array}\right) \times 0.8^{6} \times (1-0.8)^{8-6}+\left(\begin{array}{c}8\\ 7\end{array}\right) \times 0.8^{7} \times (1-0.8)^{8-7}+\left(\begin{array}{c}8\\ 8\end{array}\right) \times (1-0.8)^{8-8}\)
\(\displaystyle\Rightarrow{0.294}+{0.336}+{0.168}\)
\(\displaystyle={0.797}\)
Step 3
c) probability that the number that say they would feel secure is at most five is given by
Applying Binomial theorem
Probability of x success out of n trial is given by
\(P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right) \times p^{x} \times (1-p)^{n-x}\)
\(\displaystyle\Rightarrow{P}{\left({X}\leq{5}\right)}={1}-{P}{\left({X}{>}{5}\right)}\)
\(\displaystyle\Rightarrow{1}-{0.797}\)
\(\displaystyle={0.203}\)
Step 4
Answer
a) probability that the number that say they would feel secure is exactly five is 0.1468
b) probability that the number that say they would feel secure is more than five is 0.797
c) probability that the number that say they would feel secure is at most five is 0.203

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