Standard +0.8 This question requires students to construct a new random variable (F - 0.5M) from two independent normal distributions, apply the linear combination formula to find its mean and variance, then use standardization. While the individual steps are standard S2 techniques, the setup requires insight to recognize that 'F < 0.5M' should be rewritten as 'F - 0.5M < 0', and correctly handling the coefficient 0.5 in the variance calculation (0.5² × 55²) is a common error point. This is moderately challenging for S2 level.
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]{937c15d2-fb12-4af8-96d3-c54c81d771ba-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]{937c15d2-fb12-4af8-96d3-c54c81d771ba-06_76_1659_484_244}\\
\hfill \mbox{\textit{CAIE S2 2020 Q3 [6]}}