Question

Gouges on a spindle. A tool-and-die machine shop produces extremely high-tolerance spindles. The spindles are 18-inch slender rods used in a variety of military equipment. A piece of equipment used in the manufacture of the spindles malfunctions on occasion and places a single gouge somewhere on the spindle. However, if the spindle can be cut so that it has 14 consecutive inches without a gouge, then the spindle can be salvaged for other purposes. Assuming that the location of the gouge along the spindle is random, what is the probability that a defective spindle can be salvaged?

Short answer

The probability that a defective spindle can be salvaged is 0.4444.

Step-by-step solution

Given information

The spindle is an 18-inches slender rod. The defective spindle can be cut so that there are 14 consecutive inches without the gouge.

Let X represents the location of the gouge on the slender rod. Therefore, X is uniformly distributed between 0 inches and 18 inches.

Computing the probability of the defective spindle

The probability density function of X is:

$f\left(x\right)=\frac{1}{18};0⩽x⩽18$.

One can 14 consecutive inches without the gouge if it is present within 4 inches of either side on the slender rod.

The probability that a defective spindle can be salvaged is obtained as:

$\begin{array}{rcl}P\left(X<4\right)+P\left(X>14\right)& =& P\left(0<X<4\right)+P\left(14<X<18\right)\\ & =& \frac{4-0}{18}+\frac{18-14}{18}\\ & =& \frac{4}{18}+\frac{4}{18}\\ & =& \frac{8}{18}\\ & =& 0.4444\\ & & \end{array}$.

For the uniform distribution: $\mathit{P}\left(a<X<b\right)\mathbf{=}\frac{\mathbf{b}\mathbf{-}\mathbf{a}}{\mathbf{d}\mathbf{-}\mathbf{c}}\mathbf{;}\mathit{c}\mathbf{⩽}\mathit{a}\mathbf{<}\mathit{b}\mathbf{⩽}\mathit{d}$.

Thus, the required probability is 0.4444.