Probability Spaces and Events - Applications

The theoretical foundations of probability spaces find numerous applications in modeling real-world phenomena. We explore several important results that extend the basic framework.

Independence of Events

Definition

Events $A$ and $B$ are independent if: $P(A \cap B) = P(A) \cdot P(B)$

When $P(B) > 0$ , this is equivalent to $P(A|B) = P(A)$ , meaning knowledge of $B$ doesn't change the probability of $A$ .

For $n$ events $A_1, \ldots, A_n$ to be mutually independent, we require that for every subset $\{i_1, \ldots, i_k\}$ : $P(A_{i_1} \cap \cdots \cap A_{i_k}) = P(A_{i_1}) \cdots P(A_{i_k})$

This is stronger than pairwise independence (which only requires the condition for pairs).

Example

Pairwise but not Mutually Independent: Toss two fair coins. Define:

$A$ = "first coin is heads"
$B$ = "second coin is heads"
$C$ = "exactly one head appears"

Then $P(A) = P(B) = P(C) = 1/2$ and each pair is independent, but: $P(A \cap B \cap C) = 0 \neq P(A)P(B)P(C) = \frac{1}{8}$

Borel-Cantelli Lemmas

These fundamental results characterize the probability that infinitely many events occur.

Theorem

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events. If $\sum_{n=1}^{\infty} P(A_n) < \infty$ , then: $P(\limsup_{n \to \infty} A_n) = P(A_n \text{ occurs infinitely often}) = 0$

Theorem

If $\{A_n\}$ are mutually independent events and $\sum_{n=1}^{\infty} P(A_n) = \infty$ , then: $P(\limsup_{n \to \infty} A_n) = 1$

These lemmas provide a zero-one law: under the appropriate conditions, infinitely many events occur with probability either 0 or 1.

Example

Random Targets: Suppose an archer hits the bullseye with probability $p_n = 1/n$ on the $n$ -th shot, with shots independent. Since: $\sum_{n=1}^{\infty} \frac{1}{n} = \infty$ the second Borel-Cantelli lemma implies the archer hits the bullseye infinitely often with probability 1.

Continuity of Probability Measures

Theorem

If $A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots$ , then: $P\left(\bigcup_{n=1}^{\infty} A_n\right) = \lim_{n \to \infty} P(A_n)$

Theorem

If $A_1 \supseteq A_2 \supseteq A_3 \supseteq \cdots$ , then: $P\left(\bigcap_{n=1}^{\infty} A_n\right) = \lim_{n \to \infty} P(A_n)$

These properties show that probability measures are continuous with respect to monotone sequences of events, a crucial property for proving limit theorems.

Remark

The concept of independence is central to probability theory. Without independence, computing probabilities becomes intractable. Most probabilistic models assume some form of independence structure, making these results applicable to countless real-world scenarios.