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 roug

a. The proportion of products that exhibit edge roughness can be calculated by taking a weighted average of the percentages based on the proportion of blades used. The calculation would be: (0.25)(0.01) + (0.60)(0.03) + (0.15)(0.05) = 0.036 or 3.6%. b. To find the probability that a paper with rough edges was slit by a new knife blade, we need to use Bayes' theorem. Let A be the event that the paper was slit with a new blade and B be the event that the paper has rough edges. Then, P(A|B) = P(B|A) * P(A) / P(B). We know that P(B|A) = 0.01 and P(A) = 0.25 from the information given. To find P(B), we can again use a weighted average: (0.25)(0.01) + (0.60)(0.03) + (0.15)(0.05) = 0.036 or 3.6%. Therefore, P(B) = 0.036. Plugging everything into Bayes' theorem, we get P(A|B) = (0.01)(0.25) / 0.036 = 0.0694 or 6.94%. c. Similar to part b, let A be the event that the paper was slit with an average or worn blade, and B be the event that the paper has rough edges. Then, P(A|B) = P(B|A) * P(A) / P(B). We know that P(B|A) = 0.03 for average blades and 0.05 for worn blades, and that P(A) = 0.60 + 0.15 = 0.75. To find P(B), we can again use a weighted average: (0.25)(0.01) + (0.60)(0.03) + (0.15)(0.05) = 0.036 or 3.6%. Therefore, P(B) = 0.036. Plugging everything into Bayes' theorem, we get P(A|B) = [(0.03)(0.60) + (0.05)(0.15)] / 0.036 = 0.2639 or 26.39%. Therefore, there's a higher probability that the paper was slit with an average or worn blade than with a new blade.