Pascal's triangle

Jomo

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Dec 30, 2014
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Hi,
I do not understand the attached video at all. As you probably see in the video below balls are being dropped from the top center. I would expect that more balls would end up in the center since after each row a ball has to go right or left so there is high change that there will be close to an even number of L and R. I am not completely lost here by any means. This is clearly pascal triangle in disguise. The position number next to each place on the table, ie on pascal triangle (at least for the places where the ball can end up at which is always on pascal triangle) represents the number of paths that will lead you there. The highest number on each row of pascal's triangle is always in the middle and tappers off as we go away from the center. Another words I would expect that where the balls finish would follow a normal distribution. But in the videos it doesn't always happen. How can this be?

Video
 

lookagain

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The latest three posters' comments under the video make sense to me to partially explain the different results.
 

Dr.Peterson

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Clearly with small balls, momentum has a significant effect; if you stop the video around 1:18 you can see definite streams of balls all going in the same direction from one peg to the next, producing a definite bias toward the ends of the distribution. Other effects, such as interaction among balls, may also play a role, but I think this is the main one. So I agree with the top comment.

What interests me most is that this effect seems to almost exactly cancel the expected binomial distribution, resulting a nearly uniform distribution. I imagine that with more rows this might not be true, just as it would not be true with fewer.

So often we make assumptions in probability, and then forget to check that they apply to a physical situation like this one. Slow things down (as with the large balls) and probability takes over from physics.

Similarly, we so often focus on the counting of different ways to get an outcome, forgetting that they need to be equally likely ways for the probability reasoning to work.
 

Jomo

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Dec 30, 2014
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For some reason on my new computer I can not read youtube comments. Very strange.

Yes, I did see initially what happened with the small balls. However towards the end they did seem to become random. I therefore would have expected, based on what happened initially, to see a few bell curves at the end but it seemed to be uniform. There is lots of physics involved in this video
 
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