Problem

A potter makes decorative tiles in two colours, red and yellow. The length, $R$ cm, of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm. The length, $Y$ cm, of the yellow tiles has the normal distribution $N(12, 0.8^2)$. The random variables $R$ and $Y$ are independent.

A red tile and a yellow tile are chosen at random.

(a) Find the probability that the yellow tile is longer than the red tile.

Taruni buys 3 red tiles and 1 yellow tile.

(b) Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile.

Stefan defines the random variable $X = aR + bY$, where $a$ and $b$ are constants. He wants to use values of $a$ and $b$ such that $X$ has a mean of 780 and minimum variance.

(c) Find the value of $a$ and the value of $b$ that Stefan should use.