Question

2. A particular professor is known for his arbitrary grading policies. Each paper receives a grade from the set {A, A−, B+, B, B−, C+} with equal probability, indepenently of other papers. How many papers do you expect to hand in before you receive each possible grade at least onc?

Answer 1

14.7

Explanation:

Total number of grades= 6

Imagine Y to be number of papers till we get all grades once. Hence

Yi= Number of papers till we get i th newer grades

Expected value of Y₆= ?

The difference between getting a new grade maybe represented as

Xi= Yi+1 - Yi

Using above equation for Y₆, we get

[Y₆]= ∑⁵i=o Xi

which means, we need to get 5 different grades from the first grade.

Number of tries to see second new grade maybe represented as

X₁= {(6-1)/6}, which, for generalization is written as Xi=geo{(6-i)/6}

Xi represents the success probability of seeing further new grade.

Expected value of Xi is inverse of parameter of geometric distribution, which is

[Xi] = 6/(6-i) = 6.{1/(6-i)}

Expected value of Y₆= [∑⁵ i=0 Xi] = ∑⁵ i=0 [Xi]

Substituting value of [Xi] in the above expression

6.∑⁵i=0 {1/(6-i)} = 6. ∑⁶i=1 (1/i)

Now solving for 6 grades

Y₆ = 6[(1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6)]

Y₆ = 6 x 2.45 = 14.7