# The Random Walk For Dummies

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4 6 MIT Udergraduate Joural of Mathematics without ever crossig to the right of The the fial step brigs the druk back ito the bar This couts the total umber of differet paths that the druk ca take give the coditio that the first retur be at the th step Summarizig, we have the followig sequece of steps: Step The druk takes oe step to the left The ext - steps The druk follows a path, which correspods to a well-formed arragemet, that restricts the druk to be either at poit or to the left of it, ad that forces him to be at after the ( )st step Step The druk s first retur visit occurs whe he takes oe step to the right ad ito the bar Hece, the umber of paths the druk ca take is simply the Catala umber c, sice the well-ordered restrictio applies oly to the middle steps We kow that the th Catala umber is equal to + ( ) So the ( )st Catala umber is simply ( ) Allowig the druk to take his first step to the right will double the total umber of paths that the druk ca take So the total is ( ) To compute the probability of first retur, we eed oly divide this quatity by the size of the total sample space The latter was see at the ed of Sectio 3 to be sice there are steps Therefore, the probability that the druk reaches the bar for the first time after steps is ( ) / So, for example, cosider the probability that the first retur is at Step = 6; this probablity is give by ( )/ 6 = Coditioal Probability ad the First Retur We ow cosider a variat of the problem studied i the previous sectio: Give that the druk is at the bar at Step, what is the probability that this is his first retur visit? This problem is like the previous oe The oly differece is that the sample space has bee reduced Istead of cosiderig all of the possible paths, we ow cosider the total umber of paths give that, i the ed, the druk will be at the bar, which he may or may ot have passed by earlier

5 The Radom Walk For Dummies 7 To fid the probability that the druk will be at the bar after steps, we use the coclusios from Sectio 3 to obtai ( ) P (G = ) = p () q () Sice p = q =, this expressio becomes ( ) / The umerator i this expressio represets the umber of paths the druk ca take provided he is at the bar at the th step This is sample space we eed for the coditioal probability From Sectio, recall that, if the druk s first retur visit is o the th step, the the umber of paths that the druk ca take is ( ) O the other had, i Sectio 3, we foud the umber of paths that the druk ca take that put him back at the bar o Step ; this umber is simply ( ) Fially we divide these two umbers, gettig ( ) / ( ) = / ( )! ()! ( )!( )!!!, which simplifies to / ( ) = ( ) = The fial simplicity is amazig! The formula is just ad is uexpectedly simple! #(steps), Refereces [] Brualdi, R, Itroductory Combiatorics, Pretice-Hall, Third Editio, 999 [] Hogg, Robert V, Itroductio to Mathematical Statistics, Fifth Editio, 995 [3] Rota, G C, Probability Theory, Prelimiary Editio, 99

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