Random Walks Question?

Maths_Degree09 · 2009-01-25 05:26:58

What is Random Walks?

Here is my question:

The question is:

Suppose that you have £20 and need £50 for a bus ride home. Your only chance to get more money is by gambling in a casino. There is only one possible bet you can make at a casino: you must bet £10, and then a fair coin is flipped. If 'head' results you win £10 ( in other words you get your original 10 back plus an additional £10), and if it's 'tails,' you lose the £10 bet. The coin has exactly the same probability of coming up heads or tails:

What is the probability that you will get the £50 you need?

Thanks in advance guys!!

Last edited by Maths_Degree09 (2009-01-25 05:48:59)

Maths_Degree09 · 2009-01-25 05:58:41

pleaseee

mathsyperson · 2009-01-25 08:35:23

In a random walk, some variable is set at a starting value and then at each step, it's randomly increased or decreased. This continues until the variable goes outside of a set interval.

In your case, the variable starts at 20, is randomly increased or decreased by 10 each time, and stops when it reaches either 0 or 50.

It's fairly advanced maths to find analytic probabilities for random walks.
(Usually, anyway. If the bus had cost £40 then a simple symmetry argument says that there's a 1/2 chance he'll afford it.)
A common method of dealing with random walks is to run a computer simulation, say, 1000 times and estimate a probability from how many times the simulation outputs a success.

Muggleton · 2009-01-25 08:56:58

In theory, the betting game could go on forever, but in practice no game can go on forever. You can't possibly spend an eternity in the casino. Besides, you'll miss your bus.

A more practical phrasing of the question might be, say, what is the probability that you have £50 within 50 bets? Assuming you only have time for 50 bets before your bus arrives.

JaneFairfax · 2009-01-25 09:10:48

£50? For a bus ride? £50??

And I thought Londons bus fares were exorbitant.

bobbym · 2009-04-12 06:41:18

Hi Maths;

This one is easy. Think of it as a gamblers ruin problem with you having 20 pounds and your opponent having 30 pounds with the betting being 10 pounds per game. The game continues until one of you runs out of money. When he runs out of money is equivalent to you winning the 30 pounds you need to get on the bus. The common sense answer is the right one. If you both had 20 pounds then obviously the chances are 1 to 1. Since he has 50% more money than you he must be the favorite. He has 3 chances out of 5 of busting you and you have 2 chances out of five of busting him. The expected number of betting rounds before someone goes busted is just 2*3=6 .

So you have a 40% chance of getting enough dough to get on that bus. This is all covered in Finite Mathematics 4 edition p336 by Mizrahi and Sullivan. I am sure it is covered in many more books.

Last edited by bobbym (2009-04-13 06:35:56)

Math Is Fun Forum

#1 2009-01-25 05:26:58

Random Walks Question?

#2 2009-01-25 05:58:41

Re: Random Walks Question?

#3 2009-01-25 08:35:23

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#4 2009-01-25 08:56:58

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#5 2009-01-25 09:10:48

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#6 2009-04-12 06:41:18

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