Q; A spinner has eight equally spaced sectors labelled from 1 to 8.

Question

1.A spinner has eight equally spaced sectors labelled from 1 to 8. In a particular game, a

player wins points equal to double the sector’s face value if a power of two is spun. For all

other spins, the player loses the face value of the spin. Is this a fair game?

Please a provide a good explanation.

Answer 1

Fair game: The expected gain is 0.

To calculate expected gain we need these things:

1) For each sector value x = 1, 2, 3, 4, 5, 6, 7, 8, the Point Value V(x), which

is the amount gained if that sector gets selected.

It is positive for wins and negative for losses.

V(1) = -1

V(2) = 4

V(3) = -3

V(4) = 8

V(5) = -5

V(6) = -6

V(7) = -7

V(8) = 16

2) For each sector value x, the probability of getting x.

It is the same for all x: P(x) = 1/8.

3) For each sector value x, the product V(x)P(x):

V(1)P(1) = -1/8

V(2)P(2) = 4/8

V(3)P(3) = -3/8

V(4)P(4) = 8/8

V(5)P(5) = -5/8

V(6)P(6) = -6/8

V(7)P(7) = -7/8

V(8)P(8) = 16/8

4) The sum of all the above products, which is 6/8.

So it is not a fair game.

By the way I think the heading xP(x) in the given table is mislabeled.

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This is my understanding of the sentence

"player wins points equal to double the sector’s face value if a power of two is spun"

(I) Among the value 1, 2, 3, 4, 5, 6, 7, 8 there are at three powers of two:

2, 4, 8.

It is possible that 1 is also meant to be a power of 2, because 1 = 2⁰.

If so, then the result would be different from what I gave you.

However, when informally speaking, people mean powers of two to be even,

which excludes 1.

(II) "if a power of two is spun" means

"if 2 or 4 or 8 is spun"

(III) "double the sector's face value" means

4 if the spun sector is 2,

8 if the spun sector is 4,

16 if the spun sector is 8.

(IV) "player wins points equal to" means

The quantities in (III) are combined with + sign.

This is as opposed to loses, which are combined with - sign.

If you really meant "xP(x)" then I do not understand what you mean by x or P(x).