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Unformatted text preview: Physics 4120 Spring 2008 Homework #2 Solutions 1) a) If the drunk is back at the lamp post, that means that the number of steps to the right and to the left are the same, and half the total number of steps. So we just evaluate equation 1.2.6 for n 1 = n 2 = N/2 and p=q=1/2: ( 29 ( 29 2 ! 2 / ! 2 1 ) ( N N return W N N = b) In order to return to the lamp post, the number of steps to the left and to the right must be equal. The total number of steps is thus twice the number of steps to the right, which must be an even number since we can only take an integer number of steps to the right or left. If N is odd, the drunk cannot be back at the lamp post. So the probability for odd N is exactly 0. 2) a) Note that we are interested in the limit when both p << 1 and n << N ln(1- p ) N-n = ( N-n ) ln(1- p ) exact N ln(1- p ) approximation 1, because n << N ln(1- p ) N-n - Np approximation 2, because p << 1 Take the exponent of both sides, and we get our answer: (1- p ) N-n e- Np b) ) 1 )...( 2 )( 1 ( )! ( ! +--- =- n N N N N n N N If n << N , then every term in this product is close to N , so we can approximate each term as N . There are n terms in this product, so n N n N N - )! ( !...
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