Does the number of trials are not related to the success rate of probabilities?
I am confused why does the second code below have higher probabilities than the first one...
Hi fa shin,
Your professor is correct. 0.25 is the analytical answer. You have 3 points, and each point is either on the top of the circle (T), or on the bottom of the circle (B). So for any single random draw you have 8 possible outcomes (TTT, TTB, TBT, BTT, BBB, BBT, BTB,TBB) of these there are only 2 possible outcomes that have all 3 points on the same side of the circle TTT and BBB, so the probability is 2/8 or 1/4 = 0.25. Of course when randomly selecting this isn't going to be exact. If you drew four times you may get TTT all four times (not just once) or you may not get it at all. But as we draw more and more we expect that overall we will get TTT or BBB roughly 25% of the time. That is what the figure is showing you, as you increase the number of draws (up to 2000 or more) your more likely to get closer and closer to 0.25, but you may not get it exactly.
Justin, how do you define the "top of the circle"? The solution should actually be invariant to rotation.
I think a better explanation is that the triangle defined by the points (A,B,C) will contain the center (O) of the circle only when not all points are in one half of the circle. This is the case when all angles (AOB), (BOC) and (COA) have the same direction (either pos or neg). Now you have three angle-directions, and there are 8 combinations, two of which indicate a situation in which O is inside the triagle (A,B,C), indicating that the points can not be in one half of the circle. Although the result is the same, this does not require to define an arbitrary "bottom" or "top" part of the circle.
Hi Jochen,
Yes, top and bottom strictly speaking are arbitrary, but in terms of how the original question was written (the first image) the user defined 0 to 180 and 0 to -180 as the 'above' and 'below' semicircles respectively. It just seemed to me to be a more natural sounding way of describing why the professor said that 0.25 should be the 'answer' the code comes to and why this should not jump up to 0.9 as happened.
See my posts regarding "Number of higher probabilities and relations with the number of trials?"
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