Standard +0.3 This is a straightforward application of linear combinations of independent normal variables. Students need to form X - (Y+5), find its mean and variance using standard formulas, then calculate a single probability using tables. While it requires understanding the concept, the execution is mechanical with no tricky setup or interpretation.
3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance \(9 \mathrm {~m} ^ { 2 }\). His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance \(12 \mathrm {~m} ^ { 2 }\). Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.
\(P(T - SU > 0)\) or \(P(T - S > 5) = 1 - \Phi\left(\frac{0-1}{\sqrt{21}}\right) = \Phi(0.2182) = 0.586\)
B1, M1, M1, M1, A1
For correct mean and variance. Can be implied if using P(T-S>5) in next part; For consideration of P(T − SU > 0); For summing their two variances; For normalising and finding correct area from their values; For correct answer
[5]
$SU \sim N(19,12)$
$P(T - SU > 0)$ or $P(T - S > 5) = 1 - \Phi\left(\frac{0-1}{\sqrt{21}}\right) = \Phi(0.2182) = 0.586$ | B1, M1, M1, M1, A1 | For correct mean and variance. Can be implied if using P(T-S>5) in next part; For consideration of P(T − SU > 0); For summing their two variances; For normalising and finding correct area from their values; For correct answer
| [5]
3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance $9 \mathrm {~m} ^ { 2 }$. His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance $12 \mathrm {~m} ^ { 2 }$. Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.
\hfill \mbox{\textit{CAIE S2 2003 Q3 [5]}}