A point is to be randomly plotted on the $$\(x\)$$-axis, where the units are measured in $$\(\mathrm{cm}\)$$. The random variable $$\(R\)$$ represents the $$\(x\)$$ coordinate of the point on the $$\(x\)$$-axis and $$\(R\)$$ is uniformly distributed over the interval $$\([-5,19]\)$$ A negative value indicates that the point is to the left of the origin and a positive value indicates that the point is to the right of the origin. (a) Find the exact probability that the point is plotted to the right of the origin. (1) (b) Find the exact probability that the point is plotted more than $$\(3.5 \mathrm{~cm}\)$$ away from the origin. (2) (c) Sketch the cumulative distribution function of $$\(R\)$$ (2) Three independent points with $$\(x\)$$ coordinates $$\(R_{1}, R_{2}\)$$ and $$\(R_{3}\)$$ are plotted on the $$\(x\)$$-axis. (d) Find the exact probability that (i) all three points are more than $$\(10 \mathrm{~cm}\)$$ from the origin (3) (ii) the point furthest from the origin is more than $$\(10 \mathrm{~cm}\)$$ from the origin. (2)

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