Let’s consider an 8×8 board with a starting position of (4, 14, 2, 11, 8) and a final position of (2, 10, 8, 18):

We know from our simulations that each round has a probability of 1/256 of occurring. If the probabilities were correct, the last position would be (5, 15, 6, 9, 3), or 4/256 to the previous positions (4/5, 5/6, 6/7, 8/8); and the first position would be (15, 16, 4, 30, 18), or 9/256. In other words, the probability of winning in this particular roulette game is 0.0005. If the probabilities were correct, the chances of winning in this particular game are 1 in 4.2, or 1/4.2 to be exact, and would not even be worth contemplating. That’s pretty poor odds, no? It’s not worth worrying about this kind of thing in the real world, however, because it’s impossible to win an entire game with these parameters. You probably won’t even be able to win the casino in question (which may be a good indicator of a game in which the probabilities are correct), and no matter how hard you try, you won’t win against them.

But wait a minute. What about the odds we calculated for roulette games in the real world? We know that if they were correct, there were four possible outcomes when the probability of a given outcome was 100%: 10, 20, 30, and 40. These are the odds of winning for a particular game. If the probabilities were correct, the odds of winning these games would have to be between about 1 in 100 and 1 in 1000 (not even close to the correct odds). If the probabilities were correct, the odds of winning would have to be below 1 in 100 or 1 in 1000 (not even close to the correct odds), and would be far less than the odds of winning in any of the earlier experiments in this series.

Wait! But don’t I have to have the numbers in the first place?

There has to be some way to determine the probabilities, otherwise you wouldn’t have made this calculation. Fortunately we have the same tools available in our computing universe to determine the probabilities for any probability distribution we care to use. These are called distributions in statistics. They can be represented by a variable of any type, and can be written as a linear function between

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