A machine shop produces 20 inch titanium rods used in a variety of applications.

Question

A machine shop produces 20 inch titanium rods used in a variety of applications. On occasion a malfunction occurs and places a single gouge somewhere on the rod. However, if the rod can be cut so that it has 14 consecutive inches without a gouge, then the rod can be salvaged for other purposes.

Assuming the location of the gouge along the rod is described by a uniform distribution, what is the probability that a gouged rod can be salvaged?

What is the probability that 4 of the next 5 gouged rods can be salvaged? You may assume that the gouged rods are independent of each other. Hint: Think of a rod being salvaged as a success.

Answer 1

14 inches without a gouge means that the rod can be salvaged. Note that the 14 inches can be from either end, so if the gouge is between 14 and 20 or 0 and 20-14 = 6, the rod can be salvaged.

Then, since the gouge is uniformly distributed over the 20 inches, there is no problem if the gouge is in 0-6 or 14-20, which has probability (6+6)/20 = 3/5 or .6

Then, the probability that 4 of 5 can be salvaged is either

C(5,4).6^4(1-.6)^1 = 5 * .1296 * .4 = .2592 or

binom.dist(4,5,3/5,0) = .2592