Basic Theorems

Definition: [Classical Definition of Probability]
Let $\Omega$ be the sample space of an experiment. If $\Omega$ has $N$ points that are equally likely to occur, then the probability for any event $E \subset X$ is

(1) $\begin{align*} p(E) &= \frac{|E|}{N} \end{align*}$

Theorem: For any set $E \in \mathcal{F}, p(E^c) = 1 - p(E).$

Theorem: If $E \subset F$ , then

(2) $\begin{align*} p(F - E) = p(F \cap E^c) = p(F) - p(E). \end{align*}$

Theorem:

(3) $\begin{align*} p(E\cup F) &= p(E) + p(F) - p(E\cap F) \end{align*}$

Example: Suppose that out of 400 adults,

300 bike or swim or do both
160 swim
120 swim and bike

What is the probability that an adult select at random from this group bikes?

Solution:
Let $A$ be the event that a person swims and $B$ be teh event that they bike; then $p(A \cup B) = \frac{300}{400}, p(A) = \frac{160}{400},$ and $p(A \cap B) = \frac{120}{400}$ , since

$\begin{align*} p(A \cup B) &= p(A) + p(B) - p(A \cap B) \\ p(B) &= p(A \cup B) + p(A \cap B) - p(A) \\ &= \frac{300}{400} + \frac{120}{400} - \frac{160}{400} = \frac{260}{400} = 0.65 \end{align*}$

Theorem: [Inclusion-Exclusion Principle]

$\begin{align*} p\left( \bigcup_{i = 1}^{n} A_i \right) =& \sum_{i = 1}^{n}p(A_i) - \sum_{i = 1}^{n-1}\sum_{j = i+1}^{n} p (A_i \cap A_j) \\ &+ \sum_{i = 1}^{n-2}\sum_{j=i+1}^{n-1}\sum_{k=j+1}^{n} p(A_i \cap A_j \cap A_k) - ...\\ &+ (-1)^{n-1}p(A_1 \cap A_2 \cap ... \cap A_n) \end{align*}$

Theorem:

(4) $\begin{align*} p(E) &= p(E \cap F) + p(E \cap F^c) \end{align*}$

Example: In a community 32% of the population are male smokers and 27 % are female smokers. What percentage of the population smokes?
Solution:
Let $A$ be the event that a randomly selected person smokes, and let $B$ be the event that a person is male. By the above theorem

$\begin{align*} p(A) &= p(A \cap B) + p(A \cap B^c) = 0.32 + 0.27 = 0.59 \end{align*}$

Definition: A point is said to be randomly selected from an interval $(a,b)$ if any two subintervals of $(a,b)$ that have the same length are equally likely to include the point. The probability associated with the event that the subinterval $(\alpha, \beta)$ contains the point is defined to be $(\beta - \alpha) / (b - a).$

Definition: A point is said to be randomly selected from a finite set of points, $X$ , if each individual point has the same probability of being selected,

$\begin{align*} p(x_i) &= \frac{1}{|X|}; &x_i\in X \end{align*}$