I can’t answer that right now, because the data is not available.
If we use the same roulette wheels again, what color is the one that comes up as the red-white-red color with a 7, then the white, then gray, then green, then black? I don’t know the answer, because there is no answer. You couldn’t even tell me what the wheel of choice is that was generated by these roulette wheels.
The same holds true for our game board. The colors are always generated from this set of possibilities. Any given board is an independent distribution of random positions, which has the properties of being completely deterministic, except for a few properties. Our board is a set of numbers, and the sequence of digits in the numbers is the distribution of the values of the numbers, independent of the position (i.e., independent of the number of colors generated). The values are independent because not even one of the values has any influence other than the position it’s generated. We can also think of the sequence as a mathematical function — it comes to be the distribution of different outcomes from randomly chosen values of the distribution, independent of the position.
What about probability theory? You could derive the distribution from the set of possible outcomes in the set of possible positions. Let’s use this as example. Suppose one of the positions is green, which has a probability of 1, and the other is red, and both of them have the same probability of being at least 1. What happens if we play a roulette. We make our picks and take our actions. If we choose the green, then we get a random number between 0 and 1. If we chose the red, then we get a random number between 0 and 1. The distributions of the two numbers are independent. But what happens if we choose the opposite order? The distributions of both numbers are the same from 0 to 1, although they’re flipped.
Notice that if we chose the reverse order from the previous scenario, then we get a different distribution. You can now see how both the distribution and the set are connected. In both situations, the two outcomes are independent from each other. Even if they’re also random from each other, they are independent still.
Here’s an idea — suppose that we can create a function that’s independent of the position from which we take our action? That means that even if we took a random value from every position, we might get different values in different positions.
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