Given Data:

The probability that adults say that cashews are their favorite kind of nut is: \(\displaystyle{p}={0.43}\)

The total number of adults randomly selected is: \(\displaystyle{n}={12}\).

The probability that the cashew nuts are not their favorite kind of nut is,

\(\displaystyle{q}={1}-{p}\)

Substitute values in the above expression.

\(\displaystyle{q}={1}-{0.43}\)

\(\displaystyle={0.57}\)

The binomial expression to calculate the probability of 'x' adults who says that cashews are their favorite kind of nut is,

\(\displaystyle{P}{\left({x}\right)}=^{{{n}}}{C}_{{{x}}}{p}^{{{x}}}{q}^{{{n}-{x}}}\)

Substitute values in the above expression.

\(\displaystyle{P}{\left({x}\right)}=^{{{12}}}{C}_{{{x}}}\times{\left({0.43}\right)}^{{{x}}}\times{\left({0.57}\right)}^{{{12}-{x}}}\)

Step 2

(a) The expression to calculate the probability that exactly three adults say that the cashews are their favorite kind of nut is,

\(\displaystyle{P}{\left({3}\right)}=^{{{12}}}{C}_{{{3}}}\times{\left({0.43}\right)}^{{{3}}}\times{\left({0.57}\right)}^{{{12}-{3}}}\)

\(\displaystyle={220}\times{\left({0.43}\right)}^{{{3}}}\times{\left({0.57}\right)}^{{{9}}}\)

\(\displaystyle={0.1111}\)

Thus, the probability that exactly three adults say that the cashews are their favorite kind of nut is 0.111.

(b) The expression to calculate the probability that at least four of the adults like cashews nuts.

\(\displaystyle{P}{\left({x}\geq{4}\right)}={1}-{\left({P}{\left({0}\right)}+{P}{\left({1}\right)}+{P}{\left({2}\right)}+{P}{\left({3}\right)}\right)}\)

Substitute values in the above expression.

\(\displaystyle{P}{\left({x}\geq{4}\right)}={1}-{\left(^{\left\lbrace{12}\right\rbrace}{C}_{{{0}}}\times{\left({0.43}\right)}^{{{0}}}\times{\left({0.57}\right)}^{{{12}}}+^{{{12}}}{C}_{{{1}}}\times{0.43}\times{\left({0.57}\right)}^{{{11}}}+^{{{12}}}{C}_{{{2}}}\times{\left({0.43}\right)}^{{{2}}}\times{\left({0.43}\right)}^{{{2}}}\times{\left({0.57}\right)}^{{{10}}}+^{{{12}}}{C}_{{{3}}}\times{\left({0.43}\right)}^{{{3}}}\times{\left({0.57}\right)}^{{{9}}}\right)}\)

\(\displaystyle={1}-{\left({0.0017}+{0.0106}+{0.0441}+{0.1110}\right)}\)

\(\displaystyle={1}-{0.1674}\)

\(\displaystyle={0.8326}\)

Thus, the probability that at least four adults like cashews is0.8326.

Step 3

(c) The expression to calculate the probability that at most two adults' like cashews is,

\(\displaystyle{P}{\left({x}\le{2}\right)}={P}{\left({0}\right)}+{P}{\left({1}\right)}+{P}{\left({2}\right)}\)

Substitute values in the above expression.

\(\displaystyle{P}{\left({x}\le{2}\right)}=^{{{12}}}{C}_{{{0}}}\times{\left({0.43}\right)}^{{{0}}}\times{\left({0.57}\right)}^{{{12}}}+^{{{12}}}{C}_{{{1}}}\times{\left({0.43}\right)}\times{\left({0.57}\right)}^{{{11}}}+^{{{12}}}{C}_{{{2}}}\times{\left({0.43}\right)}^{{{2}}}\times{\left({0.57}\right)}^{{{10}}}\)

\(\displaystyle={0.0017}+{0.0106}+{0.0441}\)

\(\displaystyle={0.0564}\)

Thus, the probability that at most two adults like cashews is 0.0564.