Standard +0.3 This question requires understanding that X and 3Y are independent normal variables, finding the distribution of X - 3Y using standard formulas (mean and variance of linear combinations), then calculating a single probability. It's slightly above average difficulty due to the '3Y' component requiring variance multiplication by 9, but remains a straightforward application of A-level normal distribution theory with no novel insight required.
3 The lengths, in centimetres, of two types of insect, \(A\) and \(B\), are modelled by the random variables \(X \sim \mathrm {~N} ( 6.2,0.36 )\) and \(Y \sim \mathrm {~N} ( 2.4,0.25 )\) respectively.
Find the probability that the length of a randomly chosen type \(A\) insect is greater than the sum of the lengths of 3 randomly chosen type \(B\) insects.
3 The lengths, in centimetres, of two types of insect, $A$ and $B$, are modelled by the random variables $X \sim \mathrm {~N} ( 6.2,0.36 )$ and $Y \sim \mathrm {~N} ( 2.4,0.25 )$ respectively.
Find the probability that the length of a randomly chosen type $A$ insect is greater than the sum of the lengths of 3 randomly chosen type $B$ insects.\\
\hfill \mbox{\textit{CAIE S2 2022 Q3 [5]}}