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: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.Step-by-step explanation:We have these following probabilities:A 25% probability that the blade is newA 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[tex]P = P_{1} + P_{2} + P_{3}[/tex][tex]P_{1}[/tex] are those that are new and exhibit edge roughness. So [tex]P_{1} = 0.25*0.01 = 0.0025[/tex][tex]P_{2}[/tex] are those of average sharpness that exhibit rough edges. So [tex]P_{2} = 0.6*0.03 = 0.018[/tex][tex]P_{3}[/tex] are those that are worn and exhibit rough edges. So [tex]P_{3} = 0.15*0.05 = 0.0075[/tex]So[tex]P = P_{1} + P_{2} + P_{3} = 0.0025 + 0.018 + 0.0075 = 0.028[/tex]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[tex]P = \frac{0.0025}{0.028} = 0.0893[/tex]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 bladesFrom 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[tex]P = \frac{0.018 + 0.0075}{0.028} = 0.9107[/tex]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.