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Chapter 8: Q151E (page 452)

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

Expert verified

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

Step by step solution

01

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.

02

Computing the probability of the defective spindle

The probability density function of X is:

fx=118;0猢�x猢�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:

PX<4+PX>14=P0<X<4+P14<X<18=4-018+18-1418=418+418=818=0.4444.

For the uniform distribution: P(a<X<b)=b-ad-c;c猢�a<b猢�d.

Thus, the required probability is 0.4444.

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