Probability Analysis | Devin Alex
Devin Alex
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Probability Analysis

To truly manage Measurement Decision Risk, one must move beyond simple ratios like TUR and calculate the exact probabilities of making an incorrect decision. This requires an understanding of joint probability distributions and integration calculus.

The core problem involves two continuous random variables:

  • The True Value ($y$): The actual, unknowable quantity being measured.
  • The Measured Value ($y_m$): The result obtained from the measurement system, which includes measurement error.

The Joint Probability Density Function

The foundation of PFA and PFR calculations is the joint Probability Density Function (PDF), denoted as $f(y, y_m)$. This function describes the simultaneous probability of the true value being $y$ while the measured value is $y_m$.

By Bayes' theorem, the joint PDF can be expressed as the product of two distinct distributions:

$f(y, y_m) = f(y_m | y) \cdot f(y)$
  • The Measurement Distribution $f(y_m | y)$: This is the PDF of the measured value given a specific true value. It characterizes the measurement uncertainty and is almost always modeled as a Normal (Gaussian) distribution centered on the true value, with a variance equal to the standard measurement uncertainty squared ($u^2$).
  • The Prior Distribution $f(y)$: This is the PDF of the true value itself before the measurement is taken. It characterizes the manufacturing process or the historical drift of the instrument. Determining an accurate prior distribution is often the most challenging part of rigorous decision risk analysis.

Calculating Probability of False Accept (PFA)

A False Accept occurs when the measured value is within the Acceptance Limits ($-A \le y_m \le A$), but the true value is outside the Tolerance Limits ($y < -T$ or $y > T$).

The global PFA is calculated by integrating the joint PDF over the specific regions where this condition holds true:

$PFA = \int_{-A}^{A} \left( \int_{-\infty}^{-T} f(y, y_m) dy + \int_{T}^{\infty} f(y, y_m) dy \right) dy_m$

If the prior distribution is assumed to be uniform across the tolerance interval (a highly conservative, but common assumption when process history is unknown), the integral simplifies, but still generally requires numerical methods or complex error functions (erf) to evaluate.

Calculating Probability of False Reject (PFR)

Conversely, a False Reject occurs when the true value is actually conforming (within Tolerance Limits, $-T \le y \le T$), but the measurement system produces a value outside the Acceptance Limits ($y_m < -A$ or $y_m > A$).

The global PFR is calculated using a similar double integration over the complementary regions:

$PFR = \int_{-T}^{T} \left( \int_{-\infty}^{-A} f(y, y_m) dy_m + \int_{A}^{\infty} f(y, y_m) dy_m \right) dy$

Specific Risk vs. Global Risk

The equations above calculate Global Risk - the average probability of a bad decision over a large batch of items, taking into account the prior distribution of the manufacturing process.

However, metrologists are often concerned with Specific Risk: What is the probability that this *specific* measurement, which read 10.02 V, represents an out-of-tolerance condition? Specific risk uses Bayes' theorem to construct the posterior distribution $f(y | y_m)$ and integrates that distribution outside the tolerance limits. Specific risk is highly dependent on how close the individual measurement is to the tolerance boundary.