A fixed asymmetric assignment matrix biases the observed budget the same direction for every true state it is applied to. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The same asymmetric matrix starts from a balanced state

Chapter six's matrix-quality scan fixed the true state and varied the matrix. Here the matrix stays fixed at the mild asymmetric row from this chapter's first lesson, and the true state changes instead.

ptrue=(12,12)p_{\text{true}}=\left(\frac{1}{2},\frac{1}{2}\right)
Balanced true stateThe observed budget starts from an even split.true probabilitieszero 1/2one 1/2assignmentobserved probszero 4/5one 1/5over 100 shots80 zero reads20 one reads

A fixed asymmetric matrix skews every true state the same way

Each row's observed zero probability sits above the true zero probability, because the matrix under-reports prepared one more than it under-reports prepared zero, regardless of which true state is checked.

ptrue,zpobs,zNz12458034172085910222588\begin{array}{c|c|c}p_{\text{true},z}&p_{\text{obs},z}&N_z\\\frac{1}{2}&\frac{4}{5}&80\\\frac{3}{4}&\frac{17}{20}&85\\\frac{9}{10}&\frac{22}{25}&88\\\end{array}
Skewed true stateThe middle row's observed budget already leans further toward zero.true probabilitieszero 3/4one 1/4assignmentobserved probszero 17/20one 3/20over 100 shots85 zero reads15 one reads

A nearly-certain true state still carries the same bias

Even when the true state is already 9/10 toward zero, the asymmetric matrix still pushes the observed zero count above the true count: 88 expected zero reads out of 100 shots.

2225100=88\frac{22}{25}\cdot100=88
Peaked true stateThe final row is rendered as the checked near-certain case.true probabilitieszero 9/10one 1/10assignmentobserved probszero 22/25one 3/25over 100 shots88 zero reads12 one reads