After an exam, a teacher starts checking student attempts. Based on the marks he gives, he also awards a letter grade from any of the 5 letter grades; A, B, C, D, and F. How many papers on average should he expect to check to see all 5 grades awarded? At the end, generalize, your results for k - letter grades awarded.
To solve this problem, we can use the concept of the coupon collector's problem from probability theory. The coupon collector's problem deals with collecting a complete set of unique items from a randomly shuffled set. In this case, the teacher is looking to collect all 5 letter grades.
The probability of a student getting a particular letter grade is 1/5 for each grade (assuming all 5 letter grades are equally likely). So, the probability of not getting a particular letter grade in one attempt is 4/5.
Let's consider the expected number of papers the teacher checks before collecting the first letter grade. As described above, the probability of not getting the desired grade in one attempt is 4/5. So, the expected number of attempts to get the first grade is 1 / (4/5) = 5/4.
Similarly, the expected number of attempts to get the second unique grade is 1 / (3/5) = 5/3.
In general, the expected number of attempts to get the kth unique grade is 1 / ((6-k)/5).
Now, let's calculate the expected number of attempts to collect all 5 grades:
Expected number of attempts = (5/4) + (5/3) + (5/2) + (5/1) = 1375/24 ≈ 57.29
Therefore, the teacher should expect to check around 57.29 papers on average before all 5 grades are awarded.
For the general case of k-letter grades awarded, the formula for the expected number of attempts to collect all k grades is:
Expected number of attempts = 1/((5-k+1)/5) + 1/((5-k+2)/5) + ... + 1/(5/5)
For k=5, this formula simplifies to the result we obtained earlier.
Using this formula, you can calculate the expected number of papers the teacher should expect to check for any given value of k-letter grades awarded.