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Mean Value


mathmari
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Hey!!! Could you help me at the following exercise....
Consider a gamble,with the same possibility to win or to lose.If we win,we double our property,but if we lose we halve our property.Let's consider that we begin with an amount c.Which will be the mean value of our property,if we play n times(independent repetitions of the game)???

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I don't know what specific course you're taking or how advanced it is, but if you know how to calculate the expected value at each step, you can find a recurrence relation, then use that to figure out the sequence itself. Based on that sequence, you can find a pattern and figure out the answer from there.

Edited by John
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well, i will figure this out but i need more info first..

" consider that we begin with an amount c "

is there also a & b ?

" independent repetitions of the game "

what are they,
and is there more info to describe the exercise.

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It is an exercise of probabilities from the university... And how can I calculate the expected value at each step?????

We start with an initial property value c. Do you know how to calculate the expected property value after the first game? Once you do that, you can use the new value as the input for the second game, and then use the result of that as the input for the third game, and so on. After a few steps you may notice a pattern, and that pattern will allow you to express the expected property value after n games. I'm sorry if I'm being a bit unclear. Trying not to give too much away.

 

Also, this is assuming I'm understanding the question correctly. I take the "mean value" in this case to be the expected value of the property, but maybe you mean something different.

Edited by John
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c is an amount (a constant),the money we have at the beginning of the game....

Each time we play the game,

  • if we win we double our property(for example,at the first time we would have 2c) and
  • if we lose we halve our property(at the first time we would have c/2)

 

4c

|

2c------

| |

| c

c----

| c

| |

c/2-----

|

c/4

 

red means winning the game

green means losing the game

 

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We start with an initial property value c. Do you know how to calculate the expected property value after the first game? Once you do that, you can use the new value as the input for the second game, and then use the result of that as the input for the third game, and so on. After a few steps you may notice a pattern, and that pattern will allow you to express the expected property value after n games. I'm sorry if I'm being a bit unclear. Trying not to give too much away.

 

Also, this is assuming I'm understanding the question correctly. I take the "mean value" in this case to be the expected value of the property, but maybe you mean something different.

Yes, I mean the expected value of the property...

I have made a diagramm for 4 games, and I noticed that for n games the expected value is (2nc+c/2n+n2n-2c+nc/2n-2)/2n.... Is that correct??????

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That matches my answer when n=3, which may mean you're on the right track (or that my answer's wrong, but I think it's correct :P). How did you arrive at that result?

Edited by John
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Yes, I mean the expected value of the property...

I have made a diagramm for 4 games, and I noticed that for n games the expected value is (2nc+c/2n+n2n-2c+nc/2n-2)/2n.... Is that correct??????

 

That matches my answer when n=3, which may mean you're on the right track (or that my answer's wrong, but I think it's correct tongue.png). How did you arrive at that result?

 

John, you're answer's right. I think mathmari would have a better grasp if he followed your suggestion to play a few games and find the pattern/progression of the expected value from game to game (and use it as the answer) because it greatly simplifies the entire matter. The key, which mathmari stated (although he may not have known its significance), is that each game is independent of the others.

 

Mathmari, do you see how your current method (and trust me, I also played out a few games as you did — I did it in Excel) shows all possibilities, where each game's results depend on the previous? All this work is correct mathematically, but unnecessarily complicated (except for revealing the pattern/progression). Once you see what John is hinting at, you'll appreciate it big time.

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I tried again the exercise,and found the following:

  • n=1: E=(2c+c/2)/2=5c/4=51c/22*1
  • n=2: E=(4c+2c+c/4)/4=25c/16=52c/22*2
  • n=3: E=(14c+3c/2+c/8)/8=125c/64=53c/22*3

So, playing the game n times, the expected value is E=5nc/22*n....

Is this right???

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So, playing the game n times, the expected value is E=5nc/22*n....

Is this right???

That's what I got. You may want to change the denominator from 2^(2n) to 4^n and arrange things such that the answer is a bit "simpler," but I wouldn't expect that to matter too much.

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