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When the needle is inserted, the insertion angle θ may be deflected by some angle β. We model the probability distribution of β using a normal distribution with mean 0 and standard deviation σi; for insertion or σr for direction change. For a discrete sample of deflections (β = {−2α, −α, 0, α, 2α}), we obtain the probability of each deflection by integrating the corresponding area under the normal curve.
