what is the expected number of spots observed in rolling a carefully balanced die once?

Question

This is a statistic question and the answer is 3.5 but i dont know how solve it to get that answer!! please help!

Answer 1

Just look at the definition of expectation value. The sum over all occurrences of the value of each times the probability of each.

Answer 2

Hmm ... if I understand right, this means that whatever direction a die is rolling, any time you see a face of the die, you will also see its opposite side. So the answer is the average of any side and its opposite side.

The average of all six sides is easy to compute. Just add up all the spots and divide by six:
1 + 2 + 3 + 4 + 5 + 6 = 21 ... divide by six = 3.5.

The average of any two opposite sides is even easier ... but you first have to notice that the opposite sides of the dice always add up to seven.
1 & 6 are on opposite faces,
2 & 5 are on opposite faces,
3 & 4 are on opposite faces.
So since these two always add up to seven, we divide by the number of sides we're comparing (2 opposite sides).

7 divided by 2 = 3.5.

Or yet a third way to look at it ... if the die is rolling perfectly so that you only see four faces, if you average four faces you still get 3.5.

For example, if the die was rolling perfectly so that the 3 and its opposite side, 4, are not visible ... you would still see (in order): 1, 2, 6, 5 ... 1, 2, 6, 5 ... 1, 2, 6, 5 ... repeating. (To see this, hold a die between right thumb and forefinger on the 3 and 4, and slowly turn the die so that each side faces you.)

1 + 2 + 5 + 6 = 14 ... divide by four = 3.5.