Jacob Bernoulli
Recently at work I have given a gentle introduction to combinatorics and probability to my colleagues via a specific application of probability mass function of the binomial distribution.
It may seem like I'm repeating myself here but this presentation is a much longer exposition to this topic than my older blog entry and includes:
- A bit of historical background on two important figures in the history of maths: Jacob Bernoulli and Blaise Pascal.
- An introduction to Bernoulli trials.
- A gentle introduction to combinations and binomials.
- Intuitive interpretations of multiple formulas for binomial coefficients.
- An explanation and an intuitive interpretation of the identity of the sum of binomial coefficients from 0 to n and the cardinality of the power set.
- A brief mention of other mathematical concepts or objects such as falling factorial, multinomial coefficient, probability of independent events
Finally, combining Bernoulli trials, binomial coefficients and probability of independent events we build up a formula for probability mass function of the binomial distribution, with a practical application of its usage at the end.
My main motivation for this presentation was not to teach people about this particular application, but to introduce and explain those simpler and more fundamental pieces of maths, show how formulas can have intuitive interpretations that are more memorable than just their final textbook forms and show how relatively simple and standalone pieces of maths can be combined into a more complex expression but also how this expression can be traced back and decomposed to those simple building blocks.
PDF slides are available here. I hope some of you might find it useful or interesting, though I realize that just the slides, without spoken commentary, might be a bit dry and perhaps even confusing.