3. Show that if there are 23 people in a room, the probability is less than one half that no two of them share the same birthday.

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1 N12c Natural Sciences Part IA Dr M. G. Worster Mathematics course B Examles Sheet 1 Lent erm 2005 Please communicate any errors in this sheet to Dr Worster at M.G.Worster@damt.cam.ac.uk. Note that there is a different examles sheet for course A Probability 1. wo balls are drawn (without relacement) from a box containing five blue, four green and one yellow ball. (Like-coloured balls cannot be distinguished.) Describe the samle sace of unordered outcomes and calculate the robability of each outcome. 2. A box of 100 gaskets contains 10 gaskets with tye A defects only, 5 gaskets with tye B defects only and 2 gaskets with both tyes of defect. Given that a gasket drawn at random has a tye A defect find the robability that it also has a tye B defect. 3. Show that if there are 23 eole in a room, the robability is less than one half that no two of them share the same birthday. 4. You are travelling on the Hogwart s Exress with a comanion whom you dislike. Neither of you has a valid ticket and the insector has caught you both. She is authorised to administer a secial unishment for this offence. She holds a box containing nine Bertie Botts every-flavour beans, aarently identical. Six are fruit flavoured while three of them taste of ear wax. She makes each of you, in turn, choose and immediately eat a single bean. (a) If you choose before your comanion, what is the robability that you enjoy a fruit-flavoured bean? (b) If you choose first and escae, what is the robability that your comanion also gets a fruit flavour? (c) If you choose first and taste ear wax, what is the robability that your comanion gets a fruit flavour? (d) Is it in your better interest to ersuade your comanion to choose first? 5. A weighted die gives robability of throwing either 2,3,4 or 5, robability 2 of throwing 6 and robability /2 of throwing 1. (a) Calculate. (b) Calculate x, the exected mean score after many throws of the dice. (c) In a single throw, what is the robability of obtaining a score higher than x? (d) Calculate the variance, σ 2. (e) Check the formula σ 2 = x 2 x In the National Lottery, 6 balls are drawn at random from 49 balls, numbered from 1 to 49. You ick 6 different numbers. What is the robability that your 6 numbers match those drawn? What is the robabilty that exactly r of the numbers you choose match those drawn? What is the robability that 5 numbers you choose match those drawn and your 6th number matches a bonus ball drawn after the other 6?

2 7. I arrive home after a night at the ub and attemt to oen my front door with one of the three keys in my ocket. (You may assume that exactly one key will oen the door and that if I use it I will be successful.) Let X, a random variable, be the number of tries that I will need to oen the door if I take the keys at random from my ocket, but dro any key that fails onto the ground. Let Y, another random variable, be the number of tries needed if I take the keys at random from my ocket and immediately ut back into my ocket any key that fails. Find the robability distribution for X and Y and evaluate X and Y. [It may be useful to note that 1 + 2x + 3x 2... = (1 x) 2 if x < 1. Extra: rove this result.] 8. An oaque bag contains 10 green counters and 20 red. One counter is selected at random and then relaced: green scores 1 and red scores zero. Five draws are made. (a) Use a calculator to find r, the robability of obtaining score r = {0, 1, 2, 3, 4, 5}. Check that the robabilities sum to unity. Write down the mean r and variance σ 2 of the score. (b) Calculate the robability of obtaining scores in the ranges r ± σ/2, r ± σ. (c) he Gaussian aroximation of the binomial distribution in (a) is given as P 1 (r) ex[ 9(r 5/3) 2 /20]. Sketch P 1 (r) and r. Comare your answers in (b) with those for P 1 (r). In what sense is P 1 (r) a good aroximation to r? (d) Which of your answers would be different had you not relaced the counters after each selection? (e) Which of your answers would be different had the bag contained only 1 green and 2 red counters? 9. he size s of raindros during a storm has robability density function given by f(s) = 10ds 2 (0 s 0.6), f(s) = 9d(1 s) (0.6 s 1), f(s) = 0 otherwise. where d is a constant. (a) Find the value of d and sketch the grah of this distribution. (b) Write down the most likely size of a raindro. (c) Find the mean size of a raindro. (d) Determine the robability that the size will be (i) more than 0.8; (ii) between 0.4 and he lifetime of a bulb in a traffic signal is a random variable with density f(t) = 1 (1 t 2), f(t) = 0 otherwise, where t is measured in years. What is the robability as a function of y that the failure time of a bulb is less than y years? he traffic signal contains 3 bulbs. What is the robability as a function of z that none of the bulbs have to be relaced in z years? 11. A certain disease is known to afflict one in a thousand eole. You take a medical test that is said to be 99% accurate (i.e. it gives the correct result in 99% of the cases in which it is used). What is the robability that you actually have the disease if the test indicates that you do? Discuss your assumtions imlicit in this question and your answer.

3 12. Suose that n indentical articles are laced randomly into N boxes (states). A articular configuration of this system is such that there are n s articles in state s, 1 s N. Show that the number of ways of realizing a articular configuration is W = n! N 1 n s!. [he roduct symbol is defined such that N a s = a 1 a 2... a N ] Exact differentials 13. Find which of the following differentials P dx + Qdy are exact, and try to find an integrating factor, µ(x) or µ(y), for those that are not. If they are exact, find a function f such that df = P dx + Qdy. In each case, find the general solution of the equation P dx + Qdy = 0. How is this solution related to f or µ? (i) (ii) (iii) (iv) (v) ydx + xdy ydx + x 2 dy (x + y)dx + (x y)dy (cosh x cos y + cosh y cos x)dx (sinh x sin y sinh y sin x)dy (cos x sin x)dx + (sin x + cos x)dy (vi) (xdy ydx)/(x 2 + y 2 ). 14. he enthaly of a gas is defined by H = U +, where U satisfies du = ds d. Determine a relationshi between the differentials of H, S and. Hence show that ( ) ( ) = S S. By regarding U as a function of and and considering two exressions for show that ( ) ( ) ( ) ( ) S S = 1. 2 U,

4 15. Given that find a function G such that Prove that du = ds d, dg = d Sd. ( ) S ( ) =. 16. he ressure,, can be considered as a function of the variables and or as a function of the variables and S. (i) By exressing (, S) in the form (, S(, )) evaluate ( ) ( ) ( ) S in terms of S (ii) Hence, using ds = du + d, show that ( ) ln ln [ ( ) Hint: ln ln = ( ) ln = ln S ( ) ] (iii) For one mole of an ideal gas, ( ) ( ) and ( ) S [ 1 ] ( U/ ) + 1. ( ) ( U/ ) U = C v, P = R, and P γ deends only on S, where C v, R and γ are constants. Use formula ( ) to find an exression for γ in terms of C v and R. (iv) What is the value of γ for a monatomic gas for which C v = 3 2 R? Conditional stationary values 17. he height h of each oint (x, y) of an area of land is given by h(x, y) = a(x + y) x 2 + y 2 + a 2, where a is a ositive constant. Find the locations and heights of the highest and lowest oints of the terrain, and also those along the x and y axes. Sketch a ma of the region by showing contours of constant h in the (x, y) lane..

5 18. Find using Lagrange multiliers the stationary values subject to the condition x 2 + y 2 = 1 of (i) xy 2 (ii) e xy and the oints at which they occur. Check your answers by imlementing the condition by the substitution x = cos θ, y = sin θ. 19. Show that the largest volume of any rectangular aralleleied inscribed inside the ellisoid x 2 a 2 + y2 b 2 + z2 c 2 = 1 is 8abc/ he area A of a triangle with sides a, b, c is given by A = [s(s a)(s b)(s c)] where s = 1 2 (a + b + c). Show, using Lagrange multiliers, that of all triangles of given erimeter 2s, the triangle of largest area is equilateral. Find (in terms of the erimeter) the largest ossible area of a right-angled triangle of given erimeter. * 21. Show that the maximum value r of (x 2 + y 2 + z 2 ) 1 2 subject to the conditions satisfies the quadratic equation: x 2 a 2 + y2 b 2 + z2 c 2 = 1, lx + my + nz = 0, l 2 a 2 a 2 r 2 + m2 b 2 b 2 r 2 + n2 c 2 c 2 r 2 = 0. Interret the roblem geometrically and give the geometrical significance of one of the Lagrange multilers.

6 22. Let he Boltzmann Distribution W = s=n (g s 1 + n s )! (g s 1)! n s! and the numbers n s be subject to the conditions ( ) N n s E s = E, and N n s = n. Show that if both g s and n s are assumed large, so that Stirling s aroximation (ln n! n ln n n) may be alied, then ln W, considered as a function of the n s is greatest when n s = g s [ ex β(es µ) 1] 1 ( ) where β and βµ are Lagrange multiliers. N.B. ( ) is in fact the Bose-Einstein distribution and follows from the formula ( ) which takes into account the effects of quantum indistinguishability for bosons. However you do not need to know any quantum mechanics to do this question! For fermions the formula for W turns out to be: W = s=n g s! n s!(g s n s )! Find the value of n s for which W has its greatest value subject to the same conditions. his is the Fermi-Dirac distribution. * 23. A finite container is filled with a monatomic, ideal gas of constant internal energy (total kinetic energy). Consider the degeneracy of the kinetic energy states and hence show that the number of articles n of kinetic energy E is distributed as n E 1/2 e βe α where α and β are Lagrange multiliers. Hence determine the most robable value and the exected value of the kinetic energy of each molecule.