The Physics of Mark 6

in response to MRC@Mark 6

Q: What is the physics behind the Mark 6 machine?
A:  Newtonian mechanics, i.e. F=ma and conservation of momentum.

Q: How are the initial conditions of each trail differ from each other?
A: Let's see some examples.
    六合彩 第129期
    六合彩 第  96期

   Obviously there is great difference in the position of blades of the windmill, and the way the balls fall.

Q: Can we conclude that Mark 6 is sensitive to initial conditions?
A: First we have to know what is mean by "sensitive to initial conditions". Let's think of several systems.
    1) A pencil rolling on the desk will eventually stop, no matter how you vary its initial speed
    2) 2 identical clocks are switched on at the same time, but with different initial settings in time. Although the time displayed on each clock is different, the difference in the time displayed is more or less the same.
    3) such thing would not happen if the velocity of the milk drop is slightly changed
          
Physicists only consider something like 3) as "sensitive to initial conditions", i.e. small difference in initial condition leads to great difference in final results. To measure this "sensitivity to initial conditions", it is assumed that final difference=initial difference*exp(constant*time). By repeating the experiment many times, we can get an average value of that constant. It is important to note that, since the experimental error of each trail is different, the value of constant of each trail is different and is NOT reproducible. Therefore we can only use the average value of the constant. If the average of constant>0, the we would say that the system is sensitive to initial conditions, or more simply, chaotic.

We can only judge if Mark 6 is chaotic by finding that constant, which can be obtained by tracking the positions and velocities of some balls. Maybe someone can do it by extracting these data from life broadcast of Mark 6.

Q: If we have perfect information on the initial conditions, is it possible to predict the result of each trail?
A: Yes, since the machine is completely deterministic. But the calculation will be terribly difficult.

Q: Is Mark 6 a biased game?
A: Yes. However this game is fair only if ALL balls have the SAME initial conditions, which is impossible. Why initial conditions manner is because they determines the initial velocity and the number of collisions with the "windmill". Similar games like roulette, and even slot machine (老虎機) are biased.

To minimize the bias, the game setters then lengthen the "processing time" e.g. in Mark 6, some balls may collide n times, while the other collide n+m times, where n is much larger than m.

Q: Is there really a trend in Mark 6 results?
A: Maybe some numbers really appear more frequently, because of bias or just by chance. The number of possible combination of 7 balls within 49 balls is of course 49C7=85900584, while there are only several hundred sets of data available, we cannot judge whether there is any bias. However we can almost immediately conclude that, any attempt to analyse Mark 6 results with time series must fail, unless you believe that weather/planetary motion/spiritual power or other time dependent factors would affect the results.

Exercise
1. Suggest some practical ways to introduce more bias on Mark 6.