2024 SAJC P2 Q10

A factory produces wooden toy cars. Each toy car is made up of \(1\) rectangular block and \(4\) circular blocks. The masses of the rectangular blocks have the distribution \(\mathrm{N}(100, 4^2)\) and the masses of the circular blocks have the distribution \(\mathrm{N}(15, 2^2)\). The units for mass are grams.

1. Find the probability that the total mass of a randomly chosen toy car is between \(150\) and \(170\) grams.[2]
2. The probability that the total mass of \(n\) randomly chosen circular blocks is more than \(65\) grams is at least \(0.105\). Find the least value of \(n\).[3]

3. State an assumption needed for your calculations in parts (i) and (ii).

   [1]

The circular blocks are given a coating which increases the mass of each circular block by \(10\%\). The rectangular blocks are not given any coating.

4. Find the probability that the mass of a randomly chosen rectangular block is heavier than the total mass of \(4\) randomly chosen coated circular blocks by \(40\) grams.[3]

5. A factory inspector decides to randomly choose \(5\) coated circular blocks for inspection.

   A coated circular block will pass the inspection if it has a mass of more than \(18\) grams, otherwise it will fail the inspection.\n\nFind the probability that two coated circular blocks pass the inspection.

   [3]