Galton Board: Binomial

Also called

Galton board · quincunx · bean machine · binomial distribution · Bernoulli trials · Pascal's triangle · binomial coefficients · bell curve · normal distribution · probability · random walk · law of large numbers · biased coin · skewed distribution · mean and variance · द्विपद बंटन · प्रायिकता · पास्कल त्रिभुज

Topics

• At every peg a ball falls left or right with equal chance: each peg is one Bernoulli trial with p = ½.
• After 7 rows the bin a ball lands in is the number of rights it took: a binomial random variable B(7, ½).
• The number of paths to a bin is a binomial coefficient: 1, 7, 21, 35, 35, 21, 7, 1 for the eight bins.
• Each row of Pascal's triangle counts the paths to each peg; every entry is the sum of the two above it.
• P(k rights in n rows) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ: the middle bins fill most and the edges least.
• With many balls the heights settle into the bell shape; for large n the binomial approaches the normal curve.
• If every peg favours the right (p = 0.7) the pile skews right: the mean np moves from 3.5 to 4.9.
• The mean of B(n, p) is np and its variance np(1 − p); the pile is about √(np(1 − p)) bins wide.

About this model

A GLB file of about 394 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

More in Mathematics

C

Conic Sections from a Double ConeClass 113D model

Conic Sections from a Double ConeClass 113D model

Coplanar points in an inclined planeClass 113D model

Eight-octant coordinate frameClass 113D model

Ellipsoid with principal-axis frameClass 113D model

Elliptic cone solidClass 113D model