Plinko Game and Theory of a Chaos


Plinko is one of the most exciting pricing games on the iconic television show The Price Is Right. Contestants first compete in a pricing game to win up to five round, flat disks known as Plinko chips, which they then press flat against a pegboard wherever they want, releasing them whenever they want. The Plinko game chips fall down the board one at a time, bouncing off the pegs and moving horizontally as well as vertically, until they reach the bottom and land in one of the prize (or no prize) slots.

Notably, contestants who drop a chip that happens to land in the maximum prize slot, which is always found directly in the center of the board, frequently attempt to repeat the exact same drop with whatever remaining disks they have. Despite their best efforts, and despite the fact that the disks’ initial positions may be nearly identical, the final paths the disks end up traversing are almost never identical. Surprisingly, this game is an excellent illustration of chaos theory and aids in the explanation of the second law of thermodynamics in simple terms. Here’s how it works scientifically.

The Universe is fundamentally quantum mechanical in nature, with inherent indeterminism and uncertainty. If you consider a particle such as an electron, you might wonder:

What happened to this electron?

What is the speed and direction of this electron?

And where will the electron be if I look away now and return one second later?

They’re all reasonable questions, and we’d expect definitive answers to all of them.

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But what actually happens is so bizarre that it frightens even physicists who have spent their lives studying it. When you take a measurement to determine the precise location of an electron, you become more uncertain about its momentum: how fast and in which direction it moves. When momentum is measured instead, its position becomes more uncertain. You can only predict a probability distribution for its future position because you need to know both momentum and position to predict where it will arrive with any certainty in the future. You’ll need a measurement at that point in the future to figure out where it is.

This quantum mechanical oddity, however, should not concern Plinko. Quantum physics may have inherent indeterminism and uncertainty, but Newtonian physics should be perfectly adequate for large-scale, macroscopic systems. Newtonian physics, in contrast to the quantum mechanical equations that govern reality at its most fundamental level, is completely deterministic.

Newton’s laws of motion — all of which can be derived from F = ma (force equals mass times acceleration) — state that if you know the initial conditions, such as position and momentum, you should be able to predict where your object will be and what motion it will have at any point in the future. The equation F = ma predicts what will happen a moment later, and once that moment has passed, the same equation predicts what will happen after the next moment has passed.

These rules apply to any object for which quantum effects can be ignored, and Newtonian physics tells us how that object will evolve over time.

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