4 The masses, in milligrams, of three minerals found in 1 tonne of a certain kind of rock are modelled by three independent random variables \(P , Q\) and \(R\), where \(P \sim \mathrm {~N} \left( 46,19 ^ { 2 } \right) , Q \sim \mathrm {~N} \left( 53,23 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 25,10 ^ { 2 } \right)\). The total value of the minerals found in 1 tonne of rock is modelled by the random variable \(V\), where \(V = P + Q + 2 R\). Use the model to find the probability of finding minerals with a value of at least 93 in a randomly chosen tonne of rock.

## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(V) = 46 + 53 + 2 \times 25 = 149$ | B1 | |
| $\text{Var}(V) = 19^2 + 23^2 + 4 \times 10^2$ | M1 | or $\sqrt{(19^2 + 23^2 + 4 \times 10^2)}$ |
| $= 1290$ | A1 | or $\sqrt{1290}$ or 35.9 |
| $\dfrac{93 - 149}{\sqrt{1290}}$ | M1 | With their mean and their variance |
| $= -1.559$ | A1ft | ft their mean and variance providing 3 random variables used, allow $+/-$ |
| $1 - \Phi(-1.559) = \Phi(1.559)$ | M1 | Area consistent with their mean |
| $= 0.9405$ | A1 [7] | Accept 0.940 or 0.941 or 0.94 |

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4 The masses, in milligrams, of three minerals found in 1 tonne of a certain kind of rock are modelled by three independent random variables $P , Q$ and $R$, where $P \sim \mathrm {~N} \left( 46,19 ^ { 2 } \right) , Q \sim \mathrm {~N} \left( 53,23 ^ { 2 } \right)$ and $R \sim \mathrm {~N} \left( 25,10 ^ { 2 } \right)$. The total value of the minerals found in 1 tonne of rock is modelled by the random variable $V$, where $V = P + Q + 2 R$. Use the model to find the probability of finding minerals with a value of at least 93 in a randomly chosen tonne of rock.

\hfill \mbox{\textit{CAIE S2 2010 Q4 [7]}}