Solution: Let one of the two questions "It can be judged that two options are wrong" be selected as event A, "There is a question to judge whether an option is wrong.

If the wrong pair is selected as event B and the wrong pair is selected as event C, then P(A)= 1

12

，P(B)= 1

13

，P(C)=

14

(1) Judging from the meaning of the question, you can get 50 points, that is, 10 is all right, so the probability is P=

12

×

12

×

13

×

14

=

148

; ... (5 points)

(ii) Depending on the meaning of the question, the possible values of ξ are 30, 35, 40, 45, 50,

So P(ξ=30)=

12

×

12

×

23

×

34

=

18

; ... (6 points)

P(ξ=35)=

C 12 12

×

12

×

23

×

34

+

12

×

12

×

13

×

34

+

12

×

12

×

23

×

14

=

1748

; ... (8 points)

P(ξ=40)=

12

×

12

×

23

×

34

+

C 12 12

×

12

×

13

×

34

+

C 12 12

×

12

×

23

×

14

=

1748

; ... (10)

P(ξ=45)=

12

×

12

×

13

×

34

+

12

×

12

×

23

×

14

+

C 12 12

×

12

×

13

×

14

=

748

; ... (12 points)

p(ξ= 50)= 1

12

×

12

×

13

×

14

=

148

So the mathematical expectation of ξ is: Eξ=30×

18

+(35+40)×

1748

+45×

748

+50×

148

=

455 12

..... (13 points)