(i) The continuous random variable $$\(X\)$$ is uniformly distributed over the interval $$\([a, b]\)$$ Given that $$\(\mathrm{P}(8< X< 14)=\frac{1}{5}\)$$ and $$\(\mathrm{E}(X)=11\)$$ (a) write down $$\(\mathrm{P}(X> 14)\)$$ (1) (b) find $$\(\mathrm{P}(6 X> a+b)\)$$ (4) (ii) Susie makes a strip of pasta $$\(45 \mathrm{~cm}\)$$ long. She then cuts the strip of pasta, at a randomly chosen point, into two pieces. The random variable $$\(S\)$$ is the length of the shortest piece of pasta. (a) Write down the distribution of $$\(S\)$$ (1) (b) Calculate the probability that the shortest piece of pasta is less than $$\(12 \mathrm{~cm}\)$$ long. (2) Susie makes 20 strips of pasta, all $$\(45 \mathrm{~cm}\)$$ long, and separately cuts each strip of pasta, at a randomly chosen point, into two pieces. (c) Calculate the probability that exactly 6 of the pieces of pasta are less than $$\(12 \mathrm{~cm}\)$$ long. (3)

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