Question #15610

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?

Expert's answer

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.

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.

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