Question

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?

Short answer

The probability that the exams are graded in the alphabetical order of the students who took the exam is$\frac{1}{40320}$.

Step-by-step solution

Given information

Eight statistics exams are graded by a professor.

Define factorial

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

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

Compute the number of ways to grade the exams

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.

Compute the probability

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 $\frac{1}{40320}$.