F.3 Maths x1(long question)

As A is a semi-circle, it occupies half of the area of the lucky wheel.
And for B, C, and D, they occupy 1/6 of the area of the lucky wheel respectively.

Let x and y be the marks on C and D respectively.
Expected value
= (1/2)(1) + (1/6)(3) + (1/6)x + (1/6)y
= 1/2 + 1/2 + x/6 + y/6
= 1 + (x+y)/6
For the expected value to lie between 3 and 6 inclusively,
the expected value is either 4 or 5.
∴ For the expected value = 4,
1 + (x+y)/6 = 4
x+y = 18
∴ When x = 8, y = 10.
For the expected value = 5,
1 + (x+y)/6 = 5
x+y = 24
∴ When x = 11, y = 13.

Therefore, two pairs of distinct marks on regions C and D are 8&10, 11&13, where marks on the two regions can be reversed.