2.2 Classical probability

The classical approach to probability applies to situations where the set $\mathrm{\Omega }$ is finite, $\mathrm{\Omega }=\{{\omega }_1,\mathrm{\dots },{\omega }_n\}$, and each of the $n$ sample points is equally likely. The probability of the event $A$ is defined to be

where $|A|$ is the number of sample points in $A$ and $|\mathrm{\Omega }|$ is the number of sample points in $\mathrm{\Omega }$. Note that $\mathrm{P}(\{{\omega }_i\})=1/|\mathrm{\Omega }|$ for all $i=1,\mathrm{\dots },n$, so that all sample points have equal probability associated to them.

1. a.

   $\mathrm{\Omega }=\{H,T\}$ and the two sample points are equally likely. The event of a head $A=\{H\}$ contains 1 sample point. Therefore

   $\mathrm{P}(A)={\displaystyle \frac{1}{2}}.$

2. b.

   $\mathrm{\Omega }=\{1,2,3,4,5,6\}$ and the six sample points are equally likely. The event of a prime $A=\{2,3,5\}$ contains 3 sample points. Therefore

   $\mathrm{P}(A)={\displaystyle \frac{3}{6}}={\displaystyle \frac{1}{2}}.$

3. c.

   $\mathrm{\Omega }$ contains 52 equally likely sample points. The event of a heart $A$ contains 13 sample points. Therefore

   $\mathrm{P}(A)={\displaystyle \frac{13}{52}}={\displaystyle \frac{1}{4}}.$

Classical probability thus devotes a lot of effort to counting how many sample points are in the full sample space, and how many sample points are in events of interest. This is studied in its own right in a mathematical topic called combinatorics, which is studied both in the A level syllabus and then again in MATH112. It covers topics such as:

• •

  factorials

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  permutations and combinations

• •

  numbers of ways of selecting specific hands of cards

You will be relieved to know that we will not revisit most of this material here, even though it is usually contained in standard probability textbooks. Instead we will revise just one fact, and one extremely famous example.