Question

The edge roughness of a slit paper products increases as knife blades wear. Only 1% of products slit with new blades have rough edges, 3% of products shot with blades of average sharpness exhibit roughness, and 5% of the products slit with worn blades exhibit roughness. If 25% of the blades in manufacturing are new, 60% are of average sharpness, and 15% are worn blades.

a. What is the proportion of products that exhibit edge roughness?

b. If a paper was selected at random and found that it has rough edges, what is the probability that it was slit by a new knife blades?

c. If a paper was selected at random and found that it has rough edges, what is the probability that it was slit with a blade of average sharpness blades or worn blades?

Answer 1

a) 2.8% of products exhibit edge roughness.

b) If a paper has rough edges, there is an 8.93% probability that it was slit by a new knife blades.

c) If it has rough edges, there is a 91.07% probability that it was slit with a blade of average sharpness blades or worn blades.

Explanation:

We have these following probabilities:

A 25% probability that the blade is new

A 60% probability that the blade is of average sharpness.

A 15% probability that the blade is worn.

If a blade is new, a 1% probability that it has rough edges.

If a blade is of average sharpness, a 3% probability it has rough edges.

If a blade is worn, a 5% probability it has rough edges.

a. What is the proportion of products that exhibit edge roughness?

This is

`P = P_(1) + P_(2) + P_(3)`

`P_(1)` are those that are new and exhibit edge roughness. So
`P_(1) = 0.25*0.01 = 0.0025`

`P_(2)` are those of average sharpness that exhibit rough edges. So
`P_(2) = 0.6*0.03 = 0.018`

`P_(3)` are those that are worn and exhibit rough edges. So
`P_(3) = 0.15*0.05 = 0.0075`

So

`P = P_(1) + P_(2) + P_(3) = 0.0025 + 0.018 + 0.0075 = 0.028`

2.8% of products exhibit edge roughness.

b. If a paper was selected at random and found that it has rough edges, what is the probability that it was slit by a new knife blades?

From a), we found that there is a 0.028 probability that it has rough edges.

Also, there is a 0.0025 probability that it was slit by a new knife blades and have rough edges. So

`P = (0.0025)/(0.028) = 0.0893`

If a paper has rough edges, there is an 8.93% probability that it was slit by a new knife blades.

c) If a paper was selected at random and found that it has rough edges, what is the probability that it was slit with a blade of average sharpness blades or worn blades

From a), we found that there is a 0.028 probability that it has rough edges.

Also, there is a 0.018 probability that it was sliced with a blade of average sharpness and has rough edges and an 0.0075 probability that it was slices with worn blades and has rough edges. So

`P = (0.018 + 0.0075)/(0.028) = 0.9107`

If it has rough edges, there is a 91.07% probability that it was slit with a blade of average sharpness blades or worn blades.