1. Select Binomial Demonstration from the main menu.
2. Accept the default value of 0.5, or enter a probability between 0 and 1.
3. Pressing the number keys 1 - 9 will cause a delay of from 1 /10 to 8 seconds , respectively, for each sample. If you press the 0 (zero) key, the demonstration will finish as fast as possible. Pressing the Escape key will abort the demonstration.
In this program, 64 balls descend through 6 levels before coming to a rest in one of the 7 compartments at the bottom. The computer keeps track of how many balls fall into each of the compartments.
This program is a computerized version of the mechanical model that Sir Francis Galton (1822-1911) devised to demonstrate the workings of the binomial distribution. In Galton`s model, called a quincunx, a series of balls was dropped into the top of the apparatus, and travelled through a series of levels. At each level, the ball struck a divider and had an equal chance (p=0.5) of deflecting left or right before dropping to the next level and the next divider. (To see how the computer determines which way the ball deflects, run the Binomial Spinner Demonstration.)
When p = 0.5, cumulative binomial probabilities approximate a normal shape, i.e., they are distributed symmetrically and are unimodal. When p does not equal 0.5, the binomial distribution will still be unimodal, but will be skewed. When p is less than 0.5, the distribution will be negatively skewed, i.e., the peak of the distribution will not be in the centre of the distribution, it will be on the right, with relatively fewer observations on the left of the distribution. When p is greater than 0.5, the distribution will be positively skewed (the peak will be on the left side of the distribution, with relatively fewer observations on the right).
Binomial Distribution Approximates Normal Distribution When p = .5
By default, the computer sets the probability that the ball will fall to the left at 0.5 for each divider at each level. When you accept this value, you should see that the distribution of balls into compartments is approximately normal (symmetrical and unimodal).
Binomial Distribution Is Skewed When p does not = .5
Instead of accepting the default value of .5 that the ball will fall to the left, select another value. Of course, this simultaneously changes the probability that the ball will fall to the right. The probability that the ball will fall to the right is 1-p(left). So, if p(left) = 0.3, then p(right) = 0.7; if p(left) = 0.8, then p(right) = 0.2, and so on. If you choose a value for p(left) that is less than .5, the distribution will be unimodal but not symmetrical. For values of p(left) less than 0.5, the resulting distribution will negatively skewed, i.e., there will be relatively fewer observations on the left of the distribution.
If you choose a value for p(left) that is greater than .5, will be unimodal but not symmetrical. For values of p(left) greater than 0.5, the resulting distribution will be positively skewed, i.e., there will be relatively fewer observations on the right of the distribution.
The more extreme the value you choose for p(left) (the closer the value is to 0 or 1), the more skewed the resulting distribution will be.