Probability Distributions and asymmetry

[Liquidxlax]

Homework Statement

In a scattering experiment to measure the polarization of an elementary particle, a total of N = 1000 particles was scattered from a target. of these, 670 were observed to scattered to the right and 330 to the left. Assume there is no uncertainty in N_L + N_R = 1000

a) based on the experimental estimate of the probability, what is the uncertainty in N_L and N_R?

b) The asymmetry parameter is defined as A = (N_L - N_R)/(N_L + N_R). Calculate the asymmetry and its uncertainty

c) Assume that the asymmetry has been predicted to be A= 0.4, find a) and b) with the new value.

Homework Equations

σ = sqrt(Np(1-p) where p is the probability of the outcome and σ is the deviation N is the total number of trials

The Attempt at a Solution

N = 1000 and p = 1/2 so the uncertainty σ in N_L and N_R is ±15.8

simple plug and chug as well as the asymmetry

which equals 0.34

The problem I'm having is calculating the uncertainty in the asymmetry since the problem states there is no uncertainty in N.

I was thinking that maybe it would be

(σ_R + σ_L)/1000 but then that would mean that the error in the uncertainty would not change.

for part c) the new values of N_L = 300 ±15.8 and N_R = 700 ± 15.8

 

[mighty2000]

so the probability would be 0.3 for NL and 0.7 for NR which would result in an asymmetry of 0.4. The uncertainty in the asymmetry would still be ±0.034 since the error in the uncertainty is not changing with the new values.


Your approach to part a) and b) seems correct. Part c) is a bit confusing because the problem states that the asymmetry has been predicted to be A= 0.4, which means there is no uncertainty in this value. In that case, the calculations for the asymmetry and its uncertainty would remain the same as in part b). However, if you are supposed to take into account the uncertainty in the predicted value of A, you could use the formula for error propagation to calculate the uncertainty in the asymmetry:

σA = √[(∂A/∂NL)^2σNL^2 + (∂A/∂NR)^2σNR^2]

Where ∂A/∂NL and ∂A/∂NR are the partial derivatives of A with respect to NL and NR respectively. These can be calculated using the formula for A given in the problem. This would result in a non-zero uncertainty in A, which would affect the final result for part c).

Alternatively, if you are not expected to take into account the uncertainty in the predicted value of A, then your approach of simply using the new values of NL and NR to calculate the asymmetry and its uncertainty would be correct.