# How can I get the probability of a binomial distribution

How can I get the probability of a binomial distribution if the values are between 6 and 9, and the number of trials is 19?

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Liyana Mansell
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:
$$\displaystyle{P}{\left({6}\leq{X}\leq{9}\right)}={P}{\left({X}={6}\right)}+{P}{\left({X}={7}\right)}+{P}{\left({X}={8}\right)}+{P}{\left({X}={9}\right)}$$
$$\displaystyle={\left(^{\left\lbrace{19}\right\rbrace}{C}_{{{6}}}{p}^{{{6}}}{\left({1}-{p}\right)}^{{{19}-{6}}}+^{{{19}}}{C}_{{{7}}}{p}^{{{7}}}{\left({1}-{p}\right)}^{{{19}-{7}}}+^{{{19}}}{C}_{{{8}}}{p}^{{{8}}}{\left({1}-{p}\right)}^{{{19}-{8}}}+^{{{19}}}{C}_{{{9}}}{p}^{{{9}}}{\left({1}-{p}\right)}^{{{19}-{9}}}\right)}$$