Basic probability for quantum mechanics

[athrun200]

Homework Statement

See photo 1

Homework Equations

All formula for probability.

The Attempt at a Solution

See photo 2

It seems traditional formula (involve summation) can't be used.
So how to obtain answer for b.) and c.)?

Also, I am not sure for part a.

 

[George Jones]

Is a sum always used to calculate expectation values?

 

[athrun200]

Is a sum always used to calculate expectation values?

I think that is average value.
To get expectation values, we need to have wave function.
However, it seems there is no relation between the needle and wave function

 

[George Jones]

Let me try again.

Is a sum always used when calculating average values.