Question

+ Suppose 2% of cotton fabric rolls and 3% of nylon 2-124. fabric rolls contain flaws. Of the rolls used by a manufacturer, 70% are cotton and 30% are nylon. What is the probability that a omly selected roll used by the manufacturer contains flaws? 2-125+ The edge roughness of slit paper products increases asknife blades wear. Only 1% of products slit with new blades have rough edges, 3% of products slit with blades of average sharpness exhibit roughness, and 5% of 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, what is the proportion of products that exhibit edge roughness?

Answer 1

Solution : 2-125

$\mathrm{Let\:R\:denote\:the\:event\:that\:a\:product\:exhibits\:edge\:roughness.}$

$\mathrm{N=the\:events\:that\:the\:blades\:are\:new}$

$\mathrm{A=the\:events\:that\:the\:blades\:are\:average}$

$\mathrm{and}$

$\mathrm{W=the\:events\:that\:the\:blades\:are\:worn}$

$\mathrm{Now,}$

$\mathrm{P(R\mid N)}=0.01$

$\mathrm{P(R\mid A)}=0.03$

$\mathrm{P(R\mid W)}=0.05$

$\mathrm{P(N)}=0.25$

$\mathrm{P(A)}=0.60$

$\mathrm{P(W)}=0.15$

$\mathrm{\:Proportion\:of\:products\:that\:exhibit\:edge\:roughness\:\:P(R)\:\::}$

$\mathrm{P(R)}=\mathrm{P(R\mid N)}*\mathrm{P( N)}+\mathrm{P(R\mid A)}*\mathrm{P( A)}+\mathrm{P(R\mid W)}*\mathrm{P( W)}$

$=0.01*0.25+0.03*0.60+0.05*0.15$

$=0.0025+0.018+0.0075$

$=0.028$

$\mathrm{Hence,\quad\:the\:proportion\:of\:products\:that\:exhibit\:edge\:roughness\:\:is\:\quad{0.028}}$