The Super Quick Spin: What Fraction Stagger Right Through?

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Homework Statement



Please see attached

Homework Equations





The Attempt at a Solution



So I'm trying to answer the final q - what fraction stagger right through?

The answer I get is 1/16... is this right? It just seems quite small..

My reasoning was that the probability of getting through the first is 1/4 and then if its in the state |1,x> the probability of measuring |-1,z> is also a quarter.. therefore it is 1/4 times 1/4 i.e. 1/16 right?
 

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Sorry. it should be 1/3 times 1/16 right?

so 1/48?

Thanks