I recently watched part of the (mediocre) new Chris Hardwick gameshow The Wall, which is essentially an hour long big money version of “Plinko” from The Price is Right. If you’re not familiar with Plinko, there’s a wall of pegs and a disk (on The Wall it’s a ball) is dropped from the top. Where it lands at the bottom- after changing directions numerous times when it hits various pegs- determines the prize. (Video of Plinko and some clips of the ball drop on The Wall).
Something I’ve wondered while watching Plinko but have been embarrassed to ask, but will ask now: Let’s say that a ball or disk is dropped from Slot 4 at the top and after making its way down and being routed and rerouted by the peg it lands in the $1,000 slot at the bottom. The next time it’s dropped it lands 3 slots to the left in the $50,000 slot, and the third time it lands on the $1 slot at the far right. The drop never occurs with significantly more vigorous energy- it’s pretty much the same amount of force each time.
What are the variables that will make the object land in different slots when dropped three times in a row from the same opening?
That’s the whole basis for chaos theory. Small, unnoticeable differences in the starting state lead to wildly different results. The disk isn’t perfectly round, the pins at the top are not a snug fit for it’s diameter so you’re sometimes dropping it dressed left and sometimes dressed right, The pins’ elasticity is not always the same, and so on.
This is really the important bit. The game designers, and the owners of the show, can predict, given a reasonably long time period, how much they will have to pay out. This has been the basis of fairground stalls and penny arcades for millennia.
But really, the physics of the question isn’t all that interesting, anyway. If a ball lands centered on a peg, the physics just says that there’s no inherent reason for it to roll off one way or the other, and in fact if it were perfectly balanced and there were no perturbations, it could stay that way forever. But of course, in the real world there are always perturbations, and the balance is an unstable equilibrium, such that if something, anything, moves the ball a little bit to one side, it’ll tend to fall further to that side, and so eventually, the ball ends up on one side of the peg or the other.
Now, I did use some fancy terminology there, like “unstable equilibrium”, but the basic concept is still quite understandable and intuitive to everyone: A ball centered on a peg will end up falling to one side or the other. And that’s all the physics we have. The really interesting part comes from the mathematics: Where does the ball tend to end up if it repeatedly hits pegs and falls to one side or the other? This question, it turns out, has been extensively studied in mathematics, and is a special case of what’s called the binomial distribution (which turns out to be a very good approximation to the normal or Gaussian distribution, as the number of pegs increases).
The only added issue, I think, with the game the way it is played is that the player is allowed to introduce the ball via one of a number of holes arrayed laterally over the triangular peg board. So the distribution has to take into account the random starting point. But that doesn’t make it insolvable, just changes the distribution a bit.
