# Help with 1-dimensional random walk [closed]

I would like to simulate the evolution of a bankroll of a 2 player game where you have the probability of winning p. If a player wins, the bankroll grows by 1 unit minus a fee that each player pays, let's say the fee is r. So, when a player wins a game the bankroll grows by 1-2*r, since each player pays the fee. If a player loses the game obviously the bankroll diminishes by 1 unit.

In the example below p=0.517 and the fee is such that the player wins 0.959 units. When I generate the plot I am surprised to see that the line decreases while I would expect that it would increase. Am I generating the NestList incorrectly? Thanks.

RandomWalk[n_] :=
NestList[(# + If[RandomReal[] >= 0.517, 0.959, -1]) &, 0, n]

ListPlot[RandomWalk[3000], Joined -> True


## closed as too localized by Oleksandr R., whuber, Sjoerd C. de Vries, Artes, Yves KlettMay 1 '13 at 13:08

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• Although there is nothing wrong with this question, since it rests on a simple misunderstanding, I am going to vote to close it as "too localized". – Oleksandr R. May 1 '13 at 3:52

Not a direct answer to your question, but I've found RandomChoice is generally the most "intentful" of the random functions. With RandomChoice, your code would be something like:

randomWalk[n_, p_: .517] := NestList[# + RandomChoice[{p, 1 - p} -> {.959, -1}] &, 0, n];


And for example, if you wanted a 2-to-1 ratio, you would write:

randomWalk[n_] := NestList[# + RandomChoice[{2, 1} -> {.959, -1}] &, 0, n];


For this problem, notice there is no state involved, so you can generate a random stream independently and then Fold over it:

randomWalk[n_, p_: .517] := FoldList[Plus, 0, RandomChoice[{p, 1 - p} -> {.959, -1}, n]];


And if you really want to impress the girls:

randomWalk[n_, p_: .517] := Accumulate@RandomChoice[{p, 1 - p} -> {.959, -1}, n];

• (+1) But why impress only the girls? – Jens Apr 28 '13 at 5:07

You just have to turn the inequality around:

RandomWalk[n_] :=
NestList[(# + If[RandomReal[] <= 0.517, 0.959, -1]) &, 0, n]

ListPlot[RandomWalk[3000], Joined -> True]


The probability .517 is supposed to mean that the first argument of the If gets executed. Since the interval of RandomReal goes from 0 to 1, the part of the interval below p has the (length) measure given by p.

• oh right! thanks. – dabd Apr 27 '13 at 22:52