With 6 rows and p=0.5, which bin is most likely? How many ways can a ball reach it?

Middle bin (3 rights out of 6); C(6,3) = 20 paths.

What happens to the distribution width as you increase rows from 4 to 12?

Σ = √(np(1-p)) grows as √n — distribution spreads but also gets taller.

With p=0.7 (biased peg), where does the peak shift and why?

Mean = np = 6×0.7 = 4.2 — peak shifts right toward 4 bins.

Pro 🔒~15 min

Plinko Probability

Discover the normal distribution through random processes

How it works

At each peg, a ball has equal probability of going left or right. After n rows, the number of right-bounces follows a binomial distribution. With many rows and equal probability, the distribution approximates a normal (Gaussian) curve — the Central Limit Theorem in action. Standard deviation σ = √(np(1-p)) measures spread. This underlies all statistical inference, from polls to quantum measurements.

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Step-by-step

Set the number of rows and drop balls.
Watch the histogram fill up.
Drop many balls (100+) to see the normal distribution emerge clearly.
Change the right-probability from 50% to see how skewed distributions arise.

Key formulas

$P(k) = \binom{n}{k} p^k (1-p)^{n-k}$ Binomial distribution
$\mu = np,\; \sigma = \sqrt{np(1-p)}$ Mean and standard deviation

Frequently asked questions

With 6 rows and p=0.5, which bin is most likely? How many ways can a ball reach it?: Middle bin (3 rights out of 6); C(6,3) = 20 paths.
What happens to the distribution width as you increase rows from 4 to 12?: Σ = √(np(1-p)) grows as √n — distribution spreads but also gets taller.
With p=0.7 (biased peg), where does the peak shift and why?: Mean = np = 6×0.7 = 4.2 — peak shifts right toward 4 bins.