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| gamelogic |
 Posted: Sat Apr 12, 2008 11:15 am Post subject: Mathematical game |
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Forum Freshman
Joined: 12 Apr 2008
Posts: 28
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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:
Click on the image to view it at its original size
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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| Harold14370 |
| Posted: Sat Apr 12, 2008 12:25 pm Post subject: |
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| What happens when you throw a white? Do you throw again, or lose your turn? |
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| gamelogic |
| Posted: Sat Apr 12, 2008 12:30 pm Post subject: |
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| free radical |
| Posted: Sat Apr 12, 2008 12:31 pm Post subject: |
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Joined: 24 Sep 2007
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| yes, what happens with white, otherwise the chance to advance is 1/6 not 1/4. |
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| gamelogic |
| Posted: Sat Apr 12, 2008 12:37 pm Post subject: |
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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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| Harold14370 |
| Posted: Sat Apr 12, 2008 3:53 pm Post subject: |
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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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| Harold14370 |
| Posted: Sat Apr 12, 2008 9:09 pm Post subject: |
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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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| gamelogic |
| Posted: Sun Apr 13, 2008 2:47 am Post subject: |
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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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| gamelogic |
| Posted: Sun Apr 13, 2008 2:55 am Post subject: |
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| By the way, isn't the propability the same as expected value? |
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| Harold14370 |
| Posted: Sun Apr 13, 2008 3:28 am Post subject: |
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| gamelogic wrote: |
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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| gamelogic |
| Posted: Sun Apr 13, 2008 3:34 am Post subject: |
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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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| gamelogic |
| Posted: Sun Apr 13, 2008 3:40 am Post subject: |
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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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| Harold14370 |
| Posted: Sun Apr 13, 2008 3:58 am Post subject: |
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| gamelogic wrote: |
| 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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| gamelogic |
| Posted: Sun Apr 13, 2008 4:00 am Post subject: |
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| Oké, I see that it isn't as easy I thougt. I will have it try. |
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| gamelogic |
| Posted: Sun Apr 13, 2008 6:35 am Post subject: |
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