Standard +0.8 This question requires students to construct a new random variable (F - 0.5M) from two independent normal distributions, find its mean and variance using linear combination properties, then calculate a probability. While the mechanics are standard S2 content, the algebraic setup (rearranging F < 0.5M to F - 0.5M < 0) and handling the coefficient 0.5 in the variance calculation adds moderate complexity beyond routine linear combination questions.
3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively.
Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male.
\includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-06_76_1659_484_244}
3 The masses, in kilograms, of female and male animals of a certain species have the distributions $\mathrm { N } \left( 102,27 ^ { 2 } \right)$ and $\mathrm { N } \left( 170,55 ^ { 2 } \right)$ respectively.
Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male.\\
\includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-06_76_1659_484_244}\\
\hfill \mbox{\textit{CAIE S2 2020 Q3 [6]}}