A person plays a game of throwing an unbiased die under the condition that

he could get as many rupees as the number of points obtained on the die. Find the

expectation and variance of his winning. How much should he pay to play in order that

it is a fair game?

Solution:

The probability distribution of the number of rupees won by the person is given

below :

Thus,

E X

Rs

(

)

=

+

+

+

+

+

=

1

1

6

2

1

6

3

1

6

4

1

6

5

1

6

6

1

6

7

2

and

2

1

1

1

1

1

1

91

(

)

1

4

9

16

25

36

6

6

6

6

6

6

6

E X

=

+

+

+

+

+

=

2

2

91

7

35

2.82

6

2

12

=

-

=

=

. Note that the unit of

s

2

will be (Rs)

2

.

X Rs

p X

(

)

(

)

1

2

3

4

5

6

1

6

1

6

1

6

1

6

1

6

1

6

X

1

2

3

p

(

X

)

4

20

12

20

4

20

shouldn't px be 1 2 3 4 5 6