Please let me know if you know the solution to the following problem:
What is the probability of me waking up at 10am?
What additional information should be supplied so as to determine the probability? What do you exactly mean by the probability of this event? Which kind of conditional probability will make sense?
Consider the following comment by Timothy Gowers regarding the model for calculating the probability of an event involving a pair of dice:
Rolling a pair of dice (pp. 6), Mathematics: A very short introduction © Timothy Gowers, 2002 [Source]
I find probability very confusing, for example, this old post.
Like previous post, in this post I will discuss another contribution of Jacob (Jacques) Bernoulli. The motivation for this post came from Cédric Villain’s recent TED talk. Though I am not a fan of probability theory, but this “toy”, which I am going to discuss, is really interesting. Consider following illustration from a journal’s cover:
“Galton Board” was invented by Francis Galton in 1894. It provided a remarkable way to visualize the distribution obtained by performing several Bernoulli Trials in pre-digital computer era. Bernoulli trial is the simplest possible random experiment with exactly two possible outcomes, “success” and “failure”, in which the probability of success (say, p) is the same every time the experiment is conducted. If we perform these Bernoulli trials more than one time (say, n times) we get, what we call, Binomial Distribution. We get a discrete distribution like this:
And when the number of Bernoulli trials is very large (theoretically what we would call infinite number of trials), this Binomial Distribution can be approximated to Normal Distribution, which is a continuous distribution.
The Normal Distribution is important because of the Central Limit Theorem. This theorem implies that if you have many independent variables that may be generated by all kinds of distributions, assuming that nothing too crazy happens, the aggregate of those variables will tend toward a normal distribution. This universality across different domains of science makes the normal distribution one of the centerpieces of applied mathematics and statistics.
Here is a video in which James Grime demonstrates how Galton Board can be used to visualize Normal Distribution approximation of Binomial Distribution for very large number of Bernoulli trials. The trial outcome are represented graphically as a path in the Galton board: success corresponds to a bounce to the right and failure to a bounce to the left.