Paper 2, Section II, F

Probability | Part IA, 2011

A circular island has a volcano at its central point. During an eruption, lava flows from the mouth of the volcano and covers a sector with random angle Φ\Phi (measured in radians), whose line of symmetry makes a random angle Θ\Theta with some fixed compass bearing.

The variables Θ\Theta and Φ\Phi are independent. The probability density function of Θ\Theta is constant on (0,2π)(0,2 \pi) and the probability density function of Φ\Phi is of the form A(πϕ/2)A(\pi-\phi / 2) where 0<ϕ<2π0<\phi<2 \pi, and AA is a constant.

(a) Find the value of AA. Calculate the expected value and the variance of the sector angle Φ\Phi. Explain briefly how you would simulate the random variable Φ\Phi using a uniformly distributed random variable UU.

(b) H1H_{1} and H2H_{2} are two houses on the island which are collinear with the mouth of the volcano, but on different sides of it. Find

(i) the probability that H1H_{1} is hit by the lava;

(ii) the probability that both H1H_{1} and H2H_{2} are hit by the lava;

(iii) the probability that H2H_{2} is not hit by the lava given that H1H_{1} is hit.

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