Distribuição binomial
Distribuição binomial
Plinko and the Binomial Distribution
A Bernoulli trial is an experiment that results in a success with probability $p$ and a failure with probability $1-p$. A random variable is said to have a Binomial Distribution if it is the result of recording the number of successes in n independent Bernoulli trials.
In the Applet below, we have represented repeated independent Bernoulli trials by a single ball falling through an array of pins. Each time a ball falls onto a pin, it will bounce to the right (i.e. a success) with probability $p$ or to the left (i.e. a failure) with probability $1-p$. After the ball falls through the array, it lands in a bin labeled by the corresponding number of successes.
Click on a bin to see its corresponding total and probability. Alternatively, use the left and right arrow buttons to scroll through the bins. Also displayed is a confidence interval centered on the theoretical expected bin. Bins that are included in this confidence interval are highlighted in green.
Plinko Probability
-
Plinko Probability: http://phet.colorado.edu/pt/simulation/plinko-probability
Sir Francis Galton
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 bean machine, as drawn by Sir Francis Galton
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.
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 coefficient ${}^{n}{{C}_{k}}$. 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 ${}^{n}{{C}_{k}}\,{{p}^{k}}{{(1-p)}^{n-k}}$. 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.
[embeddoc url=”https://www.acasinhadamatematica.pt/cm/recursos_materiais/alabmat/0_ficheiros/Dist-Binon.xls” height=”500px” viewer=”microsoft”]
