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