Mathematical Methods for Physics and Engineering - Matematica.NET

30.16 EXERCISES30.11 A boy is selected at random from amongst the children belonging to families withn children. It is known that he has at least two sisters. Show that the probabilitythat he has k − 1brothersis(n − 1)!(2 n−1 − n)(k − 1)!(n − k)! ,for 1 ≤ k ≤ n − 2 and zero for other values of k. Assume that boys and girls areequally likely.30.12 Villages A, B, C and D are connected by overhead telephone lines joining AB,AC, BC, BD and CD. As a result of severe gales, there is a probability p (thesame for each link) that any particular link is broken.(a) Show that the probability that a call can be made from A to B is1 − p 2 − 2p 3 +3p 4 − p 5 .(b) Show that the probability that a call can be made from D to A is1 − 2p 2 − 2p 3 +5p 4 − 2p 5 .30.13 A set of 2N + 1 rods consists of one of each integer length 1, 2,...,2N,2N +1.Three, of lengths a, b and c, are selected, of which a is the longest. By consideringthe possible values of b and c, determine the number of ways in which a nondegeneratetriangle (i.e. one of non-zero area) can be formed (i) if a is even,and (ii) if a is odd. Combine these results appropriately to determine the totalnumber of non-degenerate triangles that can be formed with the 2N + 1 rods,and hence show that the probability that such a triangle can be formed from arandom selection (without replacement) of three rods is(N − 1)(4N +1).2(4N 2 − 1)30.14 A certain marksman never misses his target, which consists of a disc of unit radiuswith centre O. The probability that any given shot will hit the target within adistance t of O is t 2 ,for0≤ t ≤ 1. The marksman fires n independendent shotsat the target, and the random variable Y is the radius of the smallest circle withcentre O that encloses all the shots. Determine the PDF for Y and hence findthe expected area of the circle.The shot that is furthest from O is now rejected and the corresponding circledetermined for the remaining n − 1 shots. Show that its expected area isn − 1n +1 π.30.15 The duration (in minutes) of a telephone call made from a public call-box is arandom variable T . The probability density function of T is⎧⎪⎨ 0 t