4 Lecture 3, January 12, 2024

Definition 4.1 (Odds) Odds in favour of an event \(A\) occurring is \[ O(A) := \frac{P(A)}{1-P(A)}. \] Odds again an event \(A\) is \[ \frac{1-P(A)}{P(A)}. \]

The range of the odds is \([0,\infty)\).

It provide a measure of the likelihood of a particular outcome to happen.

Abbreviation: ““p:q”.

Note: Probability may be defined through the odds as follow. \[\begin{align*} &O(A) := \frac{P(A)}{1-P(A)} \\ &\implies O(A) - P(A)O(A) = P(A) \\ &\implies O(A) = P(A) (1+O(A)) \\ &\implies P(A) = \frac{O(A)} {1+O(A)} \end{align*}\]

Note: In finite, equally likely sample spaces, computing probabilities amounts to counting the number of elements in a set. It will often be difficult to do this manually, so we are looking for clever counting techniques in the next chapter.

Chapter 3 Counting Techniques

Addition rule v.s. Multiplication rule

For addition rule

  • Keyword for addition rule is “OR”;
  • \(|A|\) is defined to be the size of the set, aka the cardinality of the set.
  • If \(A\) and \(B\) are disjoint (i.e. \(A\cap B = \emptyset\)), then \(|A\cup B| = |A| + |B|\).
  • \(A\cup A^c = S\) where \(A \cap A^c = \emptyset\). Thus \(|S|=|A|+|A^c|\).

For multiplication rule

  • for multiplication rule is “AND”
  • An ordered k-tuple is an ordered set of \(k\) values: \((a_1,a_2,\dots,a_k)\). If the outcomes in A can be wrttien as an ordered k-tuple where there are \(n_1\) choices for \(a_1\), \(n_2\) choices for \(a_2,\dots\) and in general \(n_i\) choices for \(a_i\), then \[ |A| = n_1n_2\cdots n_k = \prod_{i=1}^k n_i. \]

Definition 4.2 (Factorial) Given \(n\) distinct objects, there are \[ n! = n \times (n-1) \times \ldots 2 \times 1, \] different ordered arrangements of length \(n\) that can be made. Note that, we define, \(0! = 1\).

  • We pronounce \(n!\) as “n factorial”.
  • The following recursive definition is useful: \[ n! = n \cdot (n-1)! \] When working with factorials, we can often cancel terms, e.g., \[ \frac{9!}{7!} = \frac{9\cdot 8 \cdot 7\cdot 6 \cdot \dots \cdot 2 \cdot 1}{7\cdot 6 \cdot \dots \cdot 2 \cdot 1}=9\cdot 8 = 72\]

Definition 4.3 (Permutation) Given \(n\) distinct objects, a permutation of size \(k\) is an \(ordered\) subset of \(k\) of the individuals. The number of permutations of size \(k\) taken from \(n\) objects is denoted \(n^{(k)}\) and \[ n^{(k)}=n(n-1)\dots (n-k+1) =\frac{n!}{(n-k)!}. \]

The tricky part of this definition is the word “ordered”. An ordering need not be numerical, for example assigning labels like “President” and “Vice-President” has the effect of ordering the individuals.

4.1 Questions from the class

  1. Can we express the odds as \(a:b\)?

YES. For instance, the example we saw in class (or the clicker question 1), if we roll a fair six sided dice, and let our event \(A:=\{\text{# is } 5 \}\). Then the odds \(O(A)=\frac{1/6}{1/5}=\frac{1}{5}\). We can see that, there is exactly one possibility we have event \(A\), whereas there are 5 possibilities that \(A\) does not happen (i.e. the number we roll out is \(1,2,3,4,6\)). We can abbreviate it as “1:5”. For a good example of Odds, WIKI provides a good one.