Binomial Probability Distribution Tree

Math is my best friend.

I frequently work with binomial distributions, and as a visualization aid I created this tree of probability distributions for each of its states up to fourteen trials. Each node represents the beta distribution formed for a given number of successes and failures. At the top is the case with zero data and the probability is spread evenly, as expected. The distributions are shown in white, and the green cup is a positional reference. The red lines lead to the node that adds one additional success to the number of trials and the blue lines similarly lead to the node that adds one additional failure. As you'd expect, the more failures there are, the more the probability distribution crowds to the left and vice versa with increasing successes (notice they have mirror symmetry left to right). You can see the distribution becomes more concentrated as the number of data points increase (the effect is most easily observed straight down the center, where the number of successes equals the number of failures, so the mean stays constant).

Here are a couple more in a different style and varying scales. The last one has the distribution means shown in green.

Probability Rules of Thumb

It can be difficult to get an intuitive sense of how likely an event is. With that in mind, I've created the above table to give an easy way to get a feel for the numbers without having to mess around with logarithms or anything. If you have an event with a likelihood of 1 in N, you can look up the probability of it having occurred at least once and multiply N by the listed constant to get the number of attempts you'd need. For example, many feel that an event that occurs with 1 in 100 probability is very rare. To get a 5% chance of it having happened once, you only need .05*100 = 5 trials. For a 50% chance (so just as likely as not), it takes .69*100 = 69 trials, and for a 95% chance of it having happened, it takes 3*100 = 300 trials.

One caveat is that this table is only a reasonable approximation for N above about 10 or so.