Standard +0.3 This question requires recognizing that comparing Y with W+X is equivalent to finding P(Y < W+X) or P(Y - W - X < 0), then applying the linear combination property that Y - W - X ~ N(μ_Y - μ_W - μ_X, σ²_Y + σ²_W + σ²_X), and finally standardizing to use normal tables. While it involves multiple steps, this is a standard application of a core S2 technique with no conceptual surprises, making it slightly easier than average.
2 Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions \(W \sim \mathrm {~N} ( 1520,450 ) , X \sim \mathrm {~N} ( 2250,720 )\) and \(Y \sim \mathrm {~N} ( 3860,1050 )\).
Find the probability that, on a particular day, \(Y\) is less than the total of \(W\) and \(X\).
2 Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions $W \sim \mathrm {~N} ( 1520,450 ) , X \sim \mathrm {~N} ( 2250,720 )$ and $Y \sim \mathrm {~N} ( 3860,1050 )$.
Find the probability that, on a particular day, $Y$ is less than the total of $W$ and $X$.\\
\hfill \mbox{\textit{CAIE S2 2020 Q2 [5]}}