Step 1

According to the provided data, the number of trials, n is 19 and the probability of success is p.

The basic formula to obtain the probability of a binomial distribution is,

\(\displaystyle{P}{\left({X}={x}\right)}=^{{{n}}}{C}_{{{x}}}{p}^{{{x}}}{\left({1}-{p}\right)}^{{{n}-{x}}}\)

Step 2

The probability of this binomial distribution can be obtained as:

\(\displaystyle{P}{\left({6}\leq{X}\leq{9}\right)}={P}{\left({X}={6}\right)}+{P}{\left({X}={7}\right)}+{P}{\left({X}={8}\right)}+{P}{\left({X}={9}\right)}\)

\(\displaystyle={\left(^{\left\lbrace{19}\right\rbrace}{C}_{{{6}}}{p}^{{{6}}}{\left({1}-{p}\right)}^{{{19}-{6}}}+^{{{19}}}{C}_{{{7}}}{p}^{{{7}}}{\left({1}-{p}\right)}^{{{19}-{7}}}+^{{{19}}}{C}_{{{8}}}{p}^{{{8}}}{\left({1}-{p}\right)}^{{{19}-{8}}}+^{{{19}}}{C}_{{{9}}}{p}^{{{9}}}{\left({1}-{p}\right)}^{{{19}-{9}}}\right)}\)

According to the provided data, the number of trials, n is 19 and the probability of success is p.

The basic formula to obtain the probability of a binomial distribution is,

\(\displaystyle{P}{\left({X}={x}\right)}=^{{{n}}}{C}_{{{x}}}{p}^{{{x}}}{\left({1}-{p}\right)}^{{{n}-{x}}}\)

Step 2

The probability of this binomial distribution can be obtained as:

\(\displaystyle{P}{\left({6}\leq{X}\leq{9}\right)}={P}{\left({X}={6}\right)}+{P}{\left({X}={7}\right)}+{P}{\left({X}={8}\right)}+{P}{\left({X}={9}\right)}\)

\(\displaystyle={\left(^{\left\lbrace{19}\right\rbrace}{C}_{{{6}}}{p}^{{{6}}}{\left({1}-{p}\right)}^{{{19}-{6}}}+^{{{19}}}{C}_{{{7}}}{p}^{{{7}}}{\left({1}-{p}\right)}^{{{19}-{7}}}+^{{{19}}}{C}_{{{8}}}{p}^{{{8}}}{\left({1}-{p}\right)}^{{{19}-{8}}}+^{{{19}}}{C}_{{{9}}}{p}^{{{9}}}{\left({1}-{p}\right)}^{{{19}-{9}}}\right)}\)