Question

Non-destructive evaluation. Non-destructive evaluation(NDE) describes methods that quantitatively characterize materials, tissues, and structures by non-invasive means, such as X-ray computed tomography, ultrasonic, and acoustic emission. Recently, NDE was used to detect defects in steel castings (JOM,May 2005). Assume that the probability that NDE detects a “hit” (i.e., predicts a defect in a steel casting) when, in fact, a defect exists is .97. (This is often called the probability of detection.) Also assume that the probability that NDE detects a hit when, in fact, no defect exists is .005. (This is called the probability of a false call.) Past experience has shown a defect occurs once in every 100 steel castings. If NDE detects a hit for a particular steel casting, what is the probability that an actual defect exists?

Short answer

The required probability is 0.6621.

Step-by-step solution

Important formula

The Baye’s formula is

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\frac{{\mathbf{B}}_{\mathbf{i}}}{\mathbf{A}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{i}}\mathbf{⌒}\mathbf{A}\mathbf{)}}{\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{P}\mathbf{\left(}{\mathbf{B}}_{\mathbf{i}}\mathbf{\right)}\mathbf{P}\mathbf{\left(}{\displaystyle \frac{A}{B_i}}\mathbf{\right)}}{\mathbf{P}\mathbf{\left(}{\mathbf{B}}_{\mathbf{1}}\mathbf{\right)}\mathbf{P}\mathbf{\left(}\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{i}}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{B}}_{\mathbf{2}}\mathbf{\right)}\mathbf{P}\mathbf{\left(}\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{2}}}\mathbf{\right)}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{B}}_{\mathbf{k}}\mathbf{\right)}\mathbf{P}\mathbf{\left(}\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{k}}}\mathbf{\right)}} \end{gathered}$$

The probability that an actual defect exists

Now,

Let A= defected, B=hit

$$\begin{gathered} P\left(A\right)=0.01 \\ P\left(B\backslash A\right)=0.97 \\ P\left(B\backslash A^c\right)=0.005 \\ P\left(A^c\right)=1-P\left(A\right) \\ =1-0.01 \\ =0.99 \end{gathered}$$

The probability that an actual defect exists given that NDE detects a hit particular steel casting is:

$$\begin{gathered} P\left(A\backslash B\right)=\frac{P\left(A\right)P\left({\displaystyle \frac{B}{A}}\right)}{P\left(A\right)P\left({\displaystyle \frac{B}{A}}\right)+P\left(A^c\right)P\left({\displaystyle \frac{B}{A^c}}\right)} \\ =\frac{\left(0.97\right)\left(0.01\right)}{\left(0.97\right)\left(0.01\right)+\left(0.005\right)\left(0.99\right)} \\ =0.6621 \end{gathered}$$

Therefore, the probability is 0.6621.