Peg Drop

How Peg Drop works

Peg Drop is a Plinko-style peg-drop game. You drop a ball into one of five columns at the top of a board of pegs. At every row the ball meets a peg and goes left or right, and after eight rows it lands in one of thirteen slots along the bottom. Each slot is worth points. You get five balls a round, and your best round is kept in this browser.

There is no stake, no balance and nothing to win or lose but the score. The game is about choosing where to aim, and about watching chance do the rest.

The path is decided before the ball moves

There is no physics simulation. A ball's whole route is a list of left-or-right choices, one per row, each a fair coin. Where it lands is simply the column you dropped it into plus the number of times it went right. The bounce you see is drawn along that list after it has been decided, so the animation can never change the result.

The board widens by one gap per row, which is why the outermost pegs never push a ball off the edge: a ball in the last gap of a row always has a gap below it on either side. Nothing bounces off a wall, so nothing bends the odds.

Where to aim

From any column, eight fair choices give the binomial spread: the ball lands straight below where it started 27.3% of the time, one slot either side 21.9% each, and four slots out only 0.39% each. The slots are scored so that every column has the same average, about 22 points a ball. The outer columns can reach the 100-point slots at the edges, but the empty slots sit right beside them. The middle columns land more steadily among the 25-point slots. So the choice is how much risk you want, not a hunt for the one right column.

The Galton board

Switch to the Galton board to drop many balls at once from a single point at the top. The board was invented by Francis Galton in 1889, who called it a bean machine, to show how many small, independent chances add up to a predictable shape.

With twelve rows of pegs, the number of ways to reach slot k is the binomial coefficient C(12, k), out of 2¹² = 4,096 equally likely paths. The middle slot can be reached 924 ways, so it should catch 22.6% of the balls. Each end slot can be reached only one way, so it catches 1 ball in 4,096, or 0.024%. The dashed line shows those expected counts for the number of balls you release, and the bars show what actually happened.

With 100 balls the bars wander well away from the line. With 2,000 they hug it. That is the law of large numbers, and the bell shape is the binomial distribution approaching the normal curve, both visible in a few seconds.

Is it really random?

Each ball's choices come from its own seeded random number stream. The tests drop 40,000 balls and check the slot counts against the binomial with a chi-squared goodness-of-fit test. A board whose balls went right 55% of the time instead of 50% fails that test by a factor of more than a hundred, so a biased board would not go unnoticed.