In Exercises 5–36, express all probabilities as fractions.

Grading Exams Your professor has just collected eight different statistics exams. If these exams are graded in random order, what is the probability that they are graded in alphabetical order of the students who took the exam?

Eight statistics exams are graded by a professor.

Factorial is a concept used to compute the counts of arranging n units of certain elements in n places.

Mathematically,$n!=n\left(n-1\right)\left(n-2\right)...\left(1\right)$.

Let A be the event that the exams were graded in the alphabetical order of the students who took the exam.

The total number of exams to be graded is eight.

The total number of ways of arranging these eight exams is computed below:

$$\begin{gathered} 8!=8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \\ =40320 \end{gathered}$$

The number of favorable ways the exams are graded according to the alphabetical order of the students who took the exam is one.

The probability that theexams were graded in the alphabetical order of the students who took the exam is computed below:

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}} \\ =\frac{1}{40320} \end{gathered}$$

Therefore, the probability that the exams were graded in the alphabetical order of the students whotookthe exam is equal to $\frac{1}{40320}$.