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 Probability and Statistics
 optimal number of goes
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jessica
New Member

Ireland
2 Posts

Posted - 08/22/2007 :  13:57:47  Show Profile
A game has two outcomes to win or to lose, the probability of winning is 1/3 and the probability of losing is 2/3, if i win i will receive 3 units but if i lose i will lose 1 unit, I've calculated that the average rate of return on this game is 33.3%. As i play the game more it becomes more likely that i will receive my 33.3% return. My question is if i play the game 1000 times which go would give me the best bang for my buck in other words, what is the optimal number
of times i should play to get the best value for money.

Ps. My thoughts are that a graph with "probability of gething 33% return" on the y-axis(scale 0 to 1) and "Number of goes" on the x-axis would produce a sigmoid like fuction with the optimal number being at the point of inflection but how do i find that point!Any help would gratefully be appreciated:)
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royhaas
Moderator

USA
3059 Posts

Posted - 08/22/2007 :  18:13:50  Show Profile
The value of the game is 1/3. Each "go" is independent. There is also the question of how many units you have to begin with. If you have only 1, you can lose it on the first go with probability 2/3. If you have infinitely many, it doesn't matter. However, this is not a problem where there is an optimal stopping time. The only thing you can say is that you will get closer to your "average ROR" the longer you play.

Now, it is true that the number of "goes" you can expect before you win is 3. It's also true that the probability that you lose the first 2 and win on the third is 4/27.

If this is a problem that you have devised, please try to explain further what you are trying to do. You also may want to conduct a web search for "gambler's ruin". If this is a homework problem, I'm not convinced you have presented it completely, or perhaps it has not been explained to you completely.
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jessica
New Member

Ireland
2 Posts

Posted - 08/23/2007 :  08:39:00  Show Profile
The question came in three parts and was verbally stated so
please excuse my poor interpretation, i will try to provide more clarity

Part1 was to work out average rate of return, given the stated stakes and probabilities, which i worked out as 33.3%.

The second part was to draw a graph with probability of getting the 33.3% return on the y-axis and number of times played on the x-axis
This graph tend towards one the more you play and looks similar to
the wikipedia sigmoid function that is linked in my first post.

Part3 was to find the point on the graph that gives the
player the best value for money i was informed that this
is the point of inflection.

I cannot do part 3 and am unsure if i need the information from
part 1 to find the point that is why i stated the question as such in my first post thank you very much for replying, I truly appreciate it.

Edited by - jessica on 08/23/2007 08:41:06
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