Answer to Question #15610 in Statistics and Probability for abid khan

Question

Expectations:
Someone rolls a fair six-sided die and you win points equal to the number shown. What is the expected number of points after one roll? After 2 rolls? After 100 rolls?

Answer 1

1) Suppose we made one roll.
Then each resulting number from 1..6 has the
same probability 1/6, whence the expected value
E = (1/6 + 2/6 + 3/6 + 4/6 +
5/6 + 6/6) = (1+2+3+4+5+6)/6 = 3.5

2) Suppose we made two rolls.
Then
the resulting numbers are 2..12.
Notice that distribution of probabilities is
symmetric
P(2)=P(12)
P(3)=P(11)
P(4)=P(10)
P(5)=P(9)
P(6)=P(8)

Hence
the expected value is 7

100) Suppose we made 100 rolls.
Then the
resulting numbers are 100..600.
Notice that distribution of probabilities is
again
symmetric
P(100)=P(600)
P(101)=P(599)
P(102)=P(598)
...

Hence
the expected value is
(600+100)/2 = 350.