Moderate -0.3 This is a straightforward application of the standard result that the sum of independent normal variables is normal, requiring only addition of means and variances, followed by a single normal probability calculation. The arithmetic is simple and the method is direct textbook application with no problem-solving insight needed.
3 Three coats of paint are sprayed onto a surface. The thicknesses, in millimetres, of the three coats have independent distributions \(\mathrm { N } \left( 0.13,0.02 ^ { 2 } \right) , \mathrm { N } \left( 0.14,0.03 ^ { 2 } \right)\) and \(\mathrm { N } \left( 0.10,0.01 ^ { 2 } \right)\). Find the probability that, at a randomly chosen place on the surface, the total thickness of the three coats of paint is less than 0.30 millimetres.
3 Three coats of paint are sprayed onto a surface. The thicknesses, in millimetres, of the three coats have independent distributions $\mathrm { N } \left( 0.13,0.02 ^ { 2 } \right) , \mathrm { N } \left( 0.14,0.03 ^ { 2 } \right)$ and $\mathrm { N } \left( 0.10,0.01 ^ { 2 } \right)$. Find the probability that, at a randomly chosen place on the surface, the total thickness of the three coats of paint is less than 0.30 millimetres.
\hfill \mbox{\textit{CAIE S2 2011 Q3 [5]}}