Elementary probability is concerned with counting problems.
Some of these are easy: What is the probability of throwing three heads in a row with a fair coin? Answer: 1/8.
Some of them are difficult: If you enter each of the numbers 1 to 9 twice randomly into a 9 by 9 grid (with only one number permitted in each square), what is the probability that you construct a valid Sudoku puzzle with exactly one solution? Answer: no idea.
Some of them are silly: Is Park Lane landed on less often than Mayfair on a Monopoly board? Answer: Yes. This is a consequence of 7 being the most likely roll, and seven squares back from Park Lane is “Go to jail”.
Some of them are weird: Two boxes contain some white balls and some black balls, with the same total number in each. From each box we take N balls one at a time (putting them back each time, and N is bigger than three). Can you arrange the numbers of white and black balls in each box so that the probability that all white balls are drawn from the first box is equal to the probability that the balls drawn from the second box are either all white or all black? Answer: you cannot do this, but proving it requires the solution to Fermat's Last Theorem.
This course is NOT about elementary probability theory. Instead we will give a very short introduction to modern probability, which begins with Kolmogorov (1903-1987), which makes probability part of a branch of mathematics called measure theory. This allows us to make more sense of what probability really means, and allows us to make precise statements like the following:
Counting arguments can be replaced by a kind of integration to deal with infinitely many possible outcomes.
Events that might happen can have zero probability.
If I choose a real number between 0 and 1 at random, then the probability that the number is rational is zero.
The averaged outcomes of repeated random experiments (like throwing a fair coin) converge to the normal distribution.
If you throw a fair coin infinitely often, the probability that you will always throw a head is zero, but the probability that you will throw on average half head and half tails exactly is one.
The ideas in this course are an important part of mathematics, and will be useful in several areas, from Set Theory to Financial Mathematics.
“In our time” on Radio 4 included a useful discussion about the history of probability.
You don't need to buy a book for this course, but you may find it helpful to look at one of the following books, each of which covers much more ground than we will.
Chung and AitSahlia, “Elementary Probability Theory
With Stochastic Processes and an Introduction to Mathematical Finance”, Springer
Adams and Guillemin, “Measure Theory and Probability”, Birkhauser.
Rosenthal, “A first look at rigorous probability theory”, World Scientific.
