Bean machine
Encyclopedia
The bean machine, also known as the quincunx
or Galton box, is a device invented by Sir Francis Galton
to demonstrate the central limit theorem
, in particular that the normal distribution is approximate to the binomial distribution.
The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.
Overlaying Pascal's triangle
onto the pins shows the number of different paths that can be taken to get to each bin.
A large-scale working model of this device can be seen at the Museum of Science, Boston
in the Mathematica exhibit.
. If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals 
. This is the probability mass function of a binomial distribution.
According to the central limit theorem
the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large.
Quincunx
A quincunx is a geometric pattern consisting of five points arranged in a cross, that is five coplanar points, four of them forming a square or rectangle and a fifth at its center...
or Galton box, is a device invented by Sir Francis Galton
Francis Galton
Sir Francis Galton /ˈfrɑːnsɪs ˈgɔːltn̩/ FRS , cousin of Douglas Strutt Galton, half-cousin of Charles Darwin, was an English Victorian polymath: anthropologist, eugenicist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician, and statistician...
to demonstrate the central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
, in particular that the normal distribution is approximate to the binomial distribution.
The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.
Overlaying Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
onto the pins shows the number of different paths that can be taken to get to each bin.
A large-scale working model of this device can be seen at the Museum of Science, Boston
Museum of Science, Boston
The Museum of Science is a Boston, Massachusetts landmark, located in Science Park, a plot of land spanning the Charles River. Along with over 500 interactive exhibits, the Museum features a number of live presentations throughout the building every day, along with shows at the Charles Hayden...
in the Mathematica exhibit.
Distribution of the balls
If a ball bounces to the right k times on its way down (and to the left on the remaining pins) it ends up in the kth bin counting from the left. Denoting the number of rows of pins in a bean machine by n, the number of paths to the kth bin on the bottom is given by the binomial coefficientBinomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
. If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals . This is the probability mass function of a binomial distribution.According to the central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large.
Games
Several games have been developed utilizing the idea of pins changing the route of balls or other objects:- BagatelleBagatelleBagatelle is a billiards-derived indoor table game, the object of which is to get a number of balls past wooden pins into holes...
- PachinkoPachinkois a type of game originating in Japan, and used as both a form of recreational arcade game and much more frequently as a gambling device, filling a niche in gambling in Japan comparable to that of the slot machine in Western gambling. A pachinko machine resembles a vertical pinball machine, but...
- PlinkoPlinkoPlinko is a pricing game on the American television game show The Price Is Right. The game involves guessing the prices of prizes to win "Plinko chips," which are later dropped down a large bean machine-style board to determine the contestant's cash prize...
- PayazzoPayazzoPayazzo is a traditional Finnish gambling arcade game, dating back to the 1920s, when it was introduced into Finland from Germany. The object of payazzo is to flick a coin into one of the winning slots. When the attempt is successful, the machine rewards the player with a couple of coins...
- Pazzo-lolo
External links
- Plinko and the Binomial Distribution Interactive simulation
- An 8 feet (2.4 m) Probability Machine (named Sir Francis) comparing stock market returns to the randomness of the beans dropping through the quincunx pattern. from Index Funds Advisors IFA.com
- A simulation with explanations
- Another simulation from John Carroll UniversityJohn Carroll UniversityJohn Carroll University is a private, co-educational Jesuit Catholic university in University Heights, Ohio, United States, a suburb of Cleveland. The university was founded in 1886 by the Society of Jesus as Saint Ignatius College.The university was founded in 1886 by the Society of Jesus, as...
- Quincunx and its relationship to normal distribution from Math Is Fun
- Dynamical turbulent flow on the Galton board with friction
- Animations for the Bean Machine by Yihui Xie using the RR (programming language)R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
package animation