The case of two events

We say that events $A, B \subseteq \Omega$ are independent if ${\mathbb P}[A \cap B] = {\mathbb P}[A] \cdot {\mathbb P}[B]$. Observe that, if ${\mathbb P}[B] >0$, this is the same as saying that ${\mathbb P}[A \mid B] = {\mathbb P}[A].$

Exercise 95

If $A$ and $B$ are independent, then so are $A$ and $\overline{B}$.

Example 36

If we roll a(n unbiased) die, consider the two events $A$ containing the odd number outcomes ($\{1,3,5\}$), and $B$ containing the square outcomes (i.e., $\{1,4\}$). These two events are independent. To verify this, consider $A \cap B = \{1\}$, so we have ${\mathbb P}[A\cap B]=1/6$, while ${\mathbb P}[A] = 1/2$ and ${\mathbb P}[B]= 1/3$.

Notice that in the above example, until we actually calculate the probabilities according to the definition, there is nothing to suggest that the two events are independent.

Exercise 96

Assume that there exist two (different) nontrivial and independent events in the probability space. Prove that $|\Omega| \geq 4.$

Definition 8

Two events $A$ and $B$ are positively correlated if ${\mathbb P}[A \cap B] > {\mathbb P}[A] \cdot {\mathbb P}[B]$, and negatively correlated if ${\mathbb P}[A \cap B] < {\mathbb P}[A] \cdot {\mathbb P}[B]$.

Intuitively, for positively correlated events, if one event happens, it would become more likely for the other event to happen. That is, when ${\mathbb P}[B] >0$, we can write the condition as ${\mathbb P}[A \mid B] > {\mathbb P}[A]$ (conditioned on $B$, $A$ becomes more likely to happen). For negatively correlated events, the trend is the opposite.

Example 37

Toss a(n unbiased) die. Let $A$ be the event that the outcome is an even number, and $B$ is the event that the outcome is at least $4$. So we have $A = \{2,4,6\}$ and $B= \{4,5,6\}$.

${\mathbb P}[A \cap B] = 1/3 > {\mathbb P}[A] \cdot {\mathbb P}[B] = (1/2)\cdot (1/2)$

Therefore, $A$ and $B$ are positively correlated.

Exercise 97

Pick a random number $x$ uniformly from $\{1,\ldots, n\}$. Let $A$ be the event that $x$ is even and $B$ that $x$ is divisible by $3$. Determine whether $A$ and $B$ are independent, negatively correlated, or positively correlated. Does your answer depend on the value of $n$? How many cases do you need to distinguish?