We know that the negative binomial is a discrete distribution whose probability mass function is given by;

\[P(X=x)=(\begin{array}{c}x+r-1\\ x\end{array})p^{r}(1-p)^{x}, x=0,1,2,...\]

Notation: \(\displaystyle{X}{\sin{{N}}}{B}{\left({r},{p}\right)}\)

Step 2

(4) The probability of passing an exam is 0.4.

Let \(\displaystyle{X}=\) Number of students without a passing grade before getting a third student with a passing grade.

Then, \(\displaystyle{X}{\sin{{N}}}{B}{\left({r}={3},{p}={0.4}\right)}\)

In our problem then we need to find \(\displaystyle{P}{\left({X}={7}\right)}\).

We need to find the probability of the 10th student is the third student with a passing grade.

\[P(X=7)=(\begin{array}{c}7+3-1\\ 7\end{array})0.4^{3}(1-0.4)^{7}\]

\[=(\begin{array}{c}9\\7\end{array})\times 0.4^{3} \times 0.6^{7}\]

\(\displaystyle={0.0645}\)