Question

A fair six-sided die is rolled until each face is observed at least once. On the average, how many rolls of the die are needed? Hint: use mathematical expectation and geometric distribution

Answer 1

Let the expected number of trials required be X0 when no number has come yet. Let X1 be the number of expected rolls required to get all faces when one of them has already come up. Similarly Let Xi be the expected number of rolls required that we get all the faces. Therefore, we have here:

X0  = X1 + 1

X1 = (1/6)*(X1 + 1) + (5/6)(X2 + 1)
6X1 = 1 + X1 + 5X2 + 5
5X1 = 6 + 5X2
X1 = X2 + 1.2

X2 = (2/6)(X2 + 1) + (4/6)(X3 + 1)
6X2 = 2X2 + 2 + 4X3 + 4
2X2 = 3 + 2X3
X2 = X3 + 1.5

X3 = (1/2)(1 + X3 + 1 + X4)
X3 = 2 + X4

X4 = (4/6)(1 + X4) + (2/6)(1 + X5)
6X4 = 4 + 4X4 + 2 + 2X5
2X4 = 6 + 2X5
X4 = 3 + X5

X5 = (5/6)(1 + X5) + (1/6)*1
6X5 = 5 + 5X5 + 1
X5 = 6

X4 = 3 + 6 = 9
X3 = 2 + 9 = 11
X2 = 11 + 1.5 = 12.5
X1 = 12.5 + 1.2 = 13.7
X0 = 14.7

Therfore 14.7 is the required exepced number of dice throws required here.