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Thread: Mathematical game

  1. #1 Mathematical game 
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    Hi,

    I am trying to make my own math game for a school project, but i want to know what all the chances are to win this game. This is the game board with the dice:



    Rules:

    - You start on the starting line.

    - You throw the dice and if you get a color you should put your pawn on the board with the corresponding color.

    - Then you must throw the dice again and if you throw the color that's one top you, you can go one stage further but if you throw another color you put your pawn on that color from the same stage.

    - i didn't want to make this game to long, so if you throw the second time while this is your second stay on this stage then you can put your dice on the next stage on the square with the same color you just threw.

    - White sides of the dice don't count until you are on the last stage because the end is white. So the chance's are 1/4 for every color until on the last stage. There the chance is 2/6 to throw white because there are to white sides on the dice.

    Now the goal is to get to the end as fast as possible.
    6 steps is perfect and 11 steps are the maximum amount steps.

    I am now counting step by step what the chances are but could somebody help me to make a calculation for this game to get all the chances?
    I am really dazzled, because i think this involves a combination of calculations to get a right formula.
    I hope a mathgenius is able to solve this problem!
    Thanks


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  3. #2  
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    What happens when you throw a white? Do you throw again, or lose your turn?


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  4. #3  
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    Sorry, yes throw again.
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  5. #4  
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    yes, what happens with white, otherwise the chance to advance is 1/6 not 1/4.
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  6. #5  
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    Until you are on the last stage, throwing white is like you had a ghost turn so you may throw again and again until you have a color.
    This is my variant of the game because I thought that then you will have the easiest way to make calculations for this unless you have any suggestions?
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  7. #6  
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    The first stage is not even a contest, because you keep throwing until you get a color. The probability of advancing on the first try at the next stage is 1/4, and is the same for 4 stages in a row, and 1/3 for the final stage, so that would be 1/4*1/4*1/4*1/4*1/3=1/768 so that's 1 chance in 768 of making it in 6 steps.

    At the opposite end you have 3/4*3/4*3/4*3/4*2/3=162/768=0.211 or a bit more than 1 chance in 5 to need the maximum number of steps.

    Calculating the probability for the numbers of steps in between the minimum and maximum will get a little more difficult.
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    Now, let's do the probability of making it in 7 steps. That means you failed to roll the proper color at stages 2, 3, 4, 5, or 6 but succeeded in all the other stages. The probability of failure at stage 2 and success in all other stages is
    3/4*1/4*1/4*1/4*1/3, or 3/768, and this is the same for failure at stages 3, 4, and 5. The probability of failure at stage 6 with success in the previous stages is 1/4*1/4*1/4*1/4*2/3, or 2/768. Adding these probabilities we get 14/768, or 1 in 54.8.

    I'll let you figure out the rest of them.
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  9. #8  
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    Great! I think I understand this.
    But the 1/3 chance comes when you are on the last square so I think I only have to write one chance extra, isn't it?

    Could you correct me if I am wrong,
    the propability for 6 steps is then: 1/4*1/4*1/4*1/4*1/4*1/3=1/3072

    7 steps: 3/4*1*1/4*1/4*1/4*1/4*1/3=1/1024
    (1 is because after failure throw you don't have to throw the color right above you so it becomes impossible to stay more than 2 turns on one level and the maximum amount of steps you can take to finish this game is then 12.)

    8 steps: 3/4*1*3/4*1*1/4*1/4*1/4*1/3=1/341.3333

    Is this the right way to continue?
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  10. #9  
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    By the way, isn't the propability the same as expected value?
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  11. #10  
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    Quote Originally Posted by gamelogic
    Great! I think I understand this.
    But the 1/3 chance comes when you are on the last square so I think I only have to write one chance extra, isn't it?

    Could you correct me if I am wrong,
    the propability for 6 steps is then: 1/4*1/4*1/4*1/4*1/4*1/3=1/3072
    No, I don't think this is right. The first stage is a gimme (probabilty is 1) because you don't have to roll a specific color to advance. Any color will do. Or have I misunderstood the rules? We are not calculating the expected value. That will be a number between 6 and 11. It will be the average score for all possible games played.
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  12. #11  
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    yes, you are right. I thought you counted the first level also as the others, thats why i secretly made a innovation but I will keep it according to the first plan then.
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  13. #12  
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    Then 7 steps makes: 3/768 and 8 steps:
    1*3/4*1*3/4*1*1/4*1/4*1/3=3/256 then I think this is correct, I will put the calculations in a few moment online, because I saw in your explanation that stage 6 also needs his own calculation.
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  14. #13  
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    Quote Originally Posted by gamelogic
    Then 7 steps makes: 3/768
    No, you only figured out one of the ways to score a 7. There are actually 5 different ways: failure at stage 2, failure at stage 3, stage 4, stage 5 or stage 6. That's why I calculated the probability of each of these and added them up. When you get to 8, it's going to get more complicated because you will have to find all the different combinations that will give you an 8: Failure at stages 2 and 3, 2 and 4, 2 and 5, 2 and 6, 3 and 4, and so on.
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  15. #14  
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    Oké, I see that it isn't as easy I thougt. I will have it try.
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  16. #15  
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    I will work like this to find the right combinations.

    Stage:
    http://spreadsheets.google.com/ar?id...&scol=0&ecol=7
    x meaning failure.
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  17. #16  
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    That link took me to Google documents and after I registered for that it just brought up a blank page.
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  18. #17  
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    With 8 steps I made this calculation:
    2,3: 1*3/4*1*3/4*1*1/4*1/4*1/3
    2,4: 1*3/4*1*1/4*3/4*1*1/4*1/3
    2,5: 1*3/4*1*1/4*1/4*3/4*1*1/3
    2,6: 1*3/4*1*1/4*1/4*1/4*2/3*1
    3,4: 1*1/4*3/4*1*3/4*1*1/4*1/3
    3,5: 1*1/4*3/4*1*1/4*3/4*1*1/3
    3,6: 1*1/4*3/4*1*1/4*1/4*2/3*1
    4,5: 1*1/4*1/4*3/4*1*3/4*1*1/3
    4,6: 1*1/4*1/4*3/4*1*1/4*2/3*1
    5,6: 1*1/4*1/4*1/4*3/4*1*2/3*1

    I will continue like this so I hope this is correct.
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  19. #18  
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    Quote Originally Posted by Harold14370
    That link took me to Google documents and after I registered for that it just brought up a blank page.
    yes, It is my fault because it is probably not possible for others to check it in the spreadsheat programm of google. but with 7 steps I got the same answer as yours so it is good. Btw, I must thank you for you support. :-D
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  20. #19  
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    8 stappen

    Verliezen op stadium:
    2,3: 1*3/4*1*3/4*1*1/4*1/4*1/3
    2,4: 1*3/4*1*1/4*3/4*1*1/4*1/3
    2,5: 1*3/4*1*1/4*1/4*3/4*1*1/3
    2,6: 1*3/4*1*1/4*1/4*1/4*2/3*1
    3,4: 1*1/4*3/4*1*3/4*1*1/4*1/3
    3,5: 1*1/4*3/4*1*1/4*3/4*1*1/3
    3,6: 1*1/4*3/4*1*1/4*1/4*2/3*1
    4,5: 1*1/4*1/4*3/4*1*3/4*1*1/3
    4,6: 1*1/4*1/4*3/4*1*1/4*2/3*1
    5,6: 1*1/4*1/4*1/4*3/4*1*2/3*1

    9 Stappen

    verliezen op stadium
    2,3,4: 1*3/4*1*3/4*1*3/4*1*1/4*1/3
    2,3,5: 1*3/4*1*3/4*1*1/4*3/4*1*1/3
    2,3,6: 1*3/4*1*3/4*1*1/4*1/4*2/3*1
    2,4,5: 1*3/4*1*1/4*3/4*1*3/4*1*1/3
    2,5,6: 1*3/4*1*1/4*1/4*3/4*1*2/3*1
    3,4,5: 1*1/4*3/4*1*3/4*1*3/4*1*1/3
    3,4,6: 1*1/4*3/4*1*3/4*1*1/4*2/3*1
    3,5,6: 1*1/4*3/4*1*1/4*3/4*1*2/3*1
    4,5,6: 1*1/4*1/4*3/4*1*3/4*1*2/3*1

    10 stappen:
    2,3,4,5: 1*3/4*1*3/4*1*3/4*1*3/4*1*1/3
    2,3,4,6: 1*3/4*1*3/4*1*3/4*1*1/4*2/3*1
    2,3,5,6: 1*3/4*1*3/4*1*1/4*3/4*1*2/3*1
    2,4,5,6: 1*3/4*1*1/4*3/4*1*3/4*1*2/3*1
    3,4,5,6: 1*1/4*3/4*1*3/4*1*3/4*1*2/3*1

    11 stappen:
    2,3,4,5,6: 1*3/4*1*3/4*1*3/4*1*3/4*1*2/3*1

    These are my calculations, I will now finish them.
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  21. #20  
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    Wonderfull, the answer that I got for 11 steps is: 0.211 and that is the same answer that you came with at the beginning. That means I am working correct like this.
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  22. #21  
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    Yes, I got them all:
    6 steps:
    1*1/4*1/4*1/4*1/4*1/3=1/768

    7 steps
    Lose on stage:
    2: 3/4*1/4*1/4*1/4*1/3=3/768
    3: 1/4*3/4*1/4*1/4*1/3=3/768
    4: 1/4*1/4*3/4*1/4*1/3=3/768
    5: 1/4*1/4*1/4*3/4*1/3=3/768
    6: 1/4*1/4*1/4*1/4*2/3=2/768_ +
    14/768

    8 steps
    Lose on stage:
    2,3: 1*3/4*1*3/4*1*1/4*1/4*1/3=3/256
    2,4: 1*3/4*1*1/4*3/4*1*1/4*1/3=3/256
    2,5: 1*3/4*1*1/4*1/4*3/4*1*1/3=3/256
    2,6: 1*3/4*1*1/4*1/4*1/4*2/3*1=1/128
    3,4: 1*1/4*3/4*1*3/4*1*1/4*1/3=3/256
    3,5: 1*1/4*3/4*1*1/4*3/4*1*1/3=3/256
    3,6: 1*1/4*3/4*1*1/4*1/4*2/3*1=1/128
    4,5: 1*1/4*1/4*3/4*1*3/4*1*1/3=3/256
    4,6: 1*1/4*1/4*3/4*1*1/4*2/3*1=1/128
    5,6: 1*1/4*1/4*1/4*3/4*1*2/3*1=1/128__+
    26/256

    9 steps
    Lose on stage:
    2,3,4: 1*3/4*1*3/4*1*3/4*1*1/4*1/3=9/256
    2,3,5: 1*3/4*1*3/4*1*1/4*3/4*1*1/3=9/256
    2,3,6: 1*3/4*1*3/4*1*1/4*1/4*2/3*1=3/128
    2,4,5: 1*3/4*1*1/4*3/4*1*3/4*1*1/3=9/256
    2,5,6: 1*3/4*1*1/4*1/4*3/4*1*2/3*1=3/128
    3,4,5: 1*1/4*3/4*1*3/4*1*3/4*1*1/3=9/256
    3,4,6: 1*1/4*3/4*1*3/4*1*1/4*2/3*1=3/128
    3,5,6: 1*1/4*3/4*1*1/4*3/4*1*2/3*1=3/128
    4,5,6: 1*1/4*1/4*3/4*1*3/4*1*2/3*1=3/128__+
    66/256

    10 steps
    Lose on stage:
    2,3,4,5: 1*3/4*1*3/4*1*3/4*1*3/4*1*1/3=1/9,5 (9,481481481)
    2,3,4,6: 1*3/4*1*3/4*1*3/4*1*1/4*2/3*1=9/128
    2,3,5,6: 1*3/4*1*3/4*1*1/4*3/4*1*2/3*1=9/128
    2,4,5,6: 1*3/4*1*1/4*3/4*1*3/4*1*2/3*1=9/128
    3,4,5,6: 1*1/4*3/4*1*3/4*1*3/4*1*2/3*1=9/128__+
    36/128
    11 steps
    Lose on stage:
    2,3,4,5,6: 1*3/4*1*3/4*1*3/4*1*3/4*1*2/3*1=27/128

    So to get the expected value I have to take the average of all this possible games played?
    Something like this: 1/768 + 14/768 + 26/256 + 66/256 + 36/128 + 27/128 = 669/768 = 223/256 ; avg is (223/256)/6=0,145 Is this the expected value?
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  23. #22  
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    The expected value is nothing more than an average. If you wanted to know the average age of a group of people, you would take the ages of everybody in the group, add them up, then divide by the number of people.

    Exampe: there are 10 people in a group. 3 of them are 16, 5 are 17, and 2 are 18. Their average age is (3*16+5*17+2*18)/10=16.9.

    You could get the same number by calculating the probability of a person in the group being 16 and multplying that by 16, then doing the same for pesons aged 17 and 18, then adding those numbers together. 16*0.3 +17*0.5 + 18*0.2 = 16.9

    Now, how would you apply that to your problem?
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  24. #23  
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    I think I understand this.
    I can say there are 5 kind of game-win categories.
    6 steps contains 1 possibility.
    7 steps contains 5 possibilities.
    8 steps contains 10 possibilities.
    9 steps contains 9 possibilities.
    10 steps contains 5 possibilities.
    11 steps contains 1 possibilitiy.

    I can try to do this with the help of you example I suppose.
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  25. #24  
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    I did this,
    (6+35+80+81+50+11)/31=8,483
    Then this should be the expected value of the outcome.
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  26. #25  
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    First, you made an error somewhere. The sum of the probabilities of all possible outcomes has to add up to 1. Yours only adds up to 0.88, so see if you can find where the error is.
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  27. #26  
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    The probability of finishing the game in 9 steps is 72*⁄*256</sub>. You left out the possibility “lose at 2, 4, 6”.

    And the probability of finishing the game in 10 steps is 99*⁄*256</sub>. The probability of losing at 2, 3, 4, 5 is 27*⁄*256</sub>.
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  28. #27  
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    Quote Originally Posted by JaneBennet
    The probability of finishing the game in 9 steps is 72 ⁄ 256</sub>. You left out the possibility “lose at 2, 4, 6”.

    And the probability of finishing the game in 10 steps is 99 ⁄ 256</sub>. The probability of losing at 2, 3, 4, 5 is 27 ⁄ 256</sub>.
    I noted at the bottom, your signature, contains a link to a formula. And mentions about some distance from the core or something.

    But wouldn't you have to calculate the centrifuge action of the earth spinning into the formula. In other words the earth is spinning and trying to throw you off at the equator more then anywhere else. If nothing else wouldn't you have to calculate that?

    Sincerely,


    William McCormick
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  29. #28  
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    Haha, this is pretty funny. I sense Harold like the wise teacher
    that let the student think for the answer and Janebennet like the
    smart student that just gives the answers, both thanks!

    The expected value that i calculated is correct I think isn't it?
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  30. #29  
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    Quote Originally Posted by gamelogic
    Haha, this is pretty funny. I sense Harold like the wise teacher
    that let the student think for the answer and Janebennet like the
    smart student that just gives the answers, both thanks!

    The expected value that i calculated is correct I think isn't it?
    No, actually I was too lazy to check your work and find the mistakes. I'm glad Jane did that.

    I don't think your method of calculating the expected value is right. Although 6 steps and 11 steps each had 1 way, they were not equally probable. What you have to do is make the corrections that Jane pointed out. Then take the probability for each score and multiply by that score, then add them up.
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  31. #30  
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    Yeah, I noticed something wrong with the following:

    Quote Originally Posted by gamelogic
    I think I understand this.
    I can say there are 5 kind of game-win categories.
    6 steps contains 1 possibility.
    7 steps contains 5 possibilities.
    8 steps contains 10 possibilities.
    9 steps contains 9 possibilities.
    10 steps contains 5 possibilities.
    11 steps contains 1 possibilitiy.
    <sup>5</sup>C<sub>3</sub> is not 9; it should be 10. That’s how I knew you must have left something out in your analysis of 9 steps. :P

    And the expected number of steps to finish the game, as Harold pointed out, should be 6×(1*⁄*768) + 7×(14*⁄*768) + … + 11×(27*⁄*128).
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  32. #31  
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    Sorry, my english comprehension isn't that good especially not with math explanations. What do you exactly do to get the last numbers?

    like 7 * (141/768)
    which score do you add with which score?
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    Sorry, my mistake. It should be 7×(14*⁄*768).

    14*⁄*768 is the probability you worked out for finishing the game in 7 steps.
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    This is then the calculation:
    6*(1/768)=18/768
    7*(14/768 )=98/768
    8*(26/256)=624/768
    9*(72/256)=1944/768
    10*(99/256) =2970/768
    11*(1/768)=11/768
    --------------------------------------------+
    5665/768=7,38
    So that is the expected value.
    Btw, if I play the game three times and I get 3 different kinds of steps to reach the end. Example: in 6 steps, 8 steps (3,4 fail) and 9 steps (4,5,6 fail) .
    If I want to make a conclusion I can say that my average is 7,667 and that I am near the expected value. And that the chance I played 3 times this game in this way is 1/768*3/256*3/128=18/50331648

    Or did I had to add them up?
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  35. #34  
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    The last one should be 11×(27*⁄*128).

    I calculated the expected value earlier and got 9<sup>2</sup>⁄<sub>3</sub>. So on average, out of every three games, you would complete two of them in 10 steps and one of them in 9 steps.
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  36. #35  
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    Quote Originally Posted by gamelogic
    This is then the calculation:
    6*(1/768)=18/768
    ?
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    Quote Originally Posted by Harold14370
    Quote Originally Posted by gamelogic
    This is then the calculation:
    6*(1/768)=18/768
    ?
    I see , I will change that.
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    Quote Originally Posted by JaneBennet
    Yeah, I noticed something wrong with the following:

    Quote Originally Posted by gamelogic
    I think I understand this.
    I can say there are 5 kind of game-win categories.
    6 steps contains 1 possibility.
    7 steps contains 5 possibilities.
    8 steps contains 10 possibilities.
    9 steps contains 9 possibilities.
    10 steps contains 5 possibilities.
    11 steps contains 1 possibilitiy.
    <sup>5</sup>C<sub>3</sub> is not 9; it should be 10. That’s how I knew you must have left something out in your analysis of 9 steps. :P

    And the expected number of steps to finish the game, as Harold pointed out, should be 6×(1*⁄*768) + 7×(14*⁄*768) + … + 11×(27*⁄*128).
    btw, how could you see that I left one possibility out by checking the amount of possibilities and why should it be 10?
    Is it maybe a propability rule, that the amount of possibilities must look like a pyramid? 1-5-10-10-5-1
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  39. #38  
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    Quote Originally Posted by gamelogic
    btw, how could you see that I left one possibility out by checking the amount of possibilities and why should it be 10?
    Is it maybe a propability rule, that the amount of possibilities must look like a pyramid? 1-5-10-10-5-1
    See this article on Combinations
    http://en.wikipedia.org/wiki/Combinations
    Jane knew that <sup>5</sup>C<sub>3</sub> is n!/k!(k-1)! = 5!/(3!2!) which equals 5*4*3*2/(3*2*2) =10.
    As it turns out it will look like a pyramid because you get the same result for n-k as for k.

    For your test, I think you should play a lot more than 3 games. Three is not much of a sample.
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    What is 5C3. I don't see any number in my information that is a 5 or a 3.
    My brains are beginning to freeze. I think learning chinese is easier than understanding math.
    Btw, I understand the most aspects of math except for propability. I always seem to misunderstand the situations.

    And how much games do you recommend? 5, 6 or even more?
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  41. #40  
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    Quote Originally Posted by gamelogic
    What is 5C3.

    5 "choose" 3. It is a combination. Similarly, 5P3 is a permutation.

    Cheers
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
    ~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
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    Quote Originally Posted by william
    Quote Originally Posted by gamelogic
    What is 5C3.

    5 "choose" 3. It is a combination. Similarly, 5P3 is a permutation.

    Cheers
    yes I see, but I don't have a total of 5 anywhere in my information.
    I wrote somewhere that there are 5 win-game situations but in fact there a 6. But still 5ncr3 gives 10, so I think you guys see something what I don't see.
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    You are finding different combinations of wins or losses in 5 steps. That's the n=5. Yes, there are 6 steps, but the first one doesn't matter, it's always a win. To score a 9 would require 3 losses, thats the k=3.
    I'd do at least 10.
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    Quote Originally Posted by Harold14370
    You are finding different combinations of wins or losses in 5 steps. That's the n=5. Yes, there are 6 steps, but the first one doesn't matter, it's always a win. To score a 9 would require 3 losses, thats the k=3.
    I'd do at least 10.
    Great, I finally understand my game. Oké, i did 10 like you said, and I see more difference in the outcome (it will also be better for my grade ), so thanks for the advice.
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    I made the conclusion about the games I played and compared them with the expected value and I also compared them with the propability and gave my comments about the results. Is that really the only thing that I can do with my results or is there something else that I forgot to use?
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    Quote Originally Posted by gamelogic
    I made the conclusion about the games I played and compared them with the expected value and I also compared them with the propability and gave my comments about the results. Is that really the only thing that I can do with my results or is there something else that I forgot to use?
    What were your experimental results and did they agree with the calculated probabilities? You could make a chart of the frequency of each score versus the calculated probablity distribution.
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    Good one, you mean something like is.
    Steps: ------- 6 ---7 --- 8 --9 -- 10 -- 11
    Frequency: 0,01 - 0,2- 1--3 --- 4 ---- 2
    (calculated for
    ten games)
    Frequency: ---1 -- 1 -- 4 -- 3 -- 0 -- 1
    ( my games)

    It seems a lot easier to talk about the results like this, thx.
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  48. #47  
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    Want to thank everybody that helped me, Thanks²
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  49. #48  
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    Quote Originally Posted by gamelogic
    Want to thank everybody that helped me, Thanks²
    You're welcome. How did it work out for you? Did you get a good grade?
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  50. #49  
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    Quote Originally Posted by Harold14370
    Quote Originally Posted by gamelogic
    Want to thank everybody that helped me, Thanks²
    You're welcome. How did it work out for you? Did you get a good grade?
    In June I'll have to present it to the teachers. Then we'll see
    the results.
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