I was trying to find the answer in some of the questions that already have answers, but I couldn't reach any conclusion with the information.
Can events that present some kind of correlation happen simultaneously, or a dependent event is also always mutually exclusive?
If not, could you give me an example?
Thanks
$\endgroup$3 Answers
$\begingroup$A very simple counterexample has the two events the same (e.g. that a coin lands on heads). Then they are maximally dependent (correlation one), but not mutually exclusive.
$\endgroup$ 4 $\begingroup$The definition of independent events says that$$ P(A\cap B) = P(A) P(B),$$that is, the probability that $A$ and $B$ both happen is the product of the individual probability that $A$ will happen and the individual probability that $B$ will happen.
In the case of two mutually exclusive events $A$ and $B$, each of which has non-zero probability, we have that $P(A)P(B) \neq 0,$but $P(A\cap B) = 0,$ so the events cannot be independent.
But there are many other ways for events to be dependent. This happens whenever $P(A\cap B)$ is anything except exactlyequal to $P(A)P(B).$
$\endgroup$ $\begingroup$I think Venn diagrams help in this situation. Draw a rectangle to represent all your outcomes (this is the sample space). Divide the rectangle in two (unequal) parts with a straight line. The events on one side will represent the event $B$ and those on the other side will represent the complement, $\overline{B}$.
In general, another event $A$ will intersect both $B$ and $\overline{B}$. If the probability of getting an outcome that lies in $A$ "depends" on whether we are allowed to select the outcome from the entire sample space or we are only allowed to select it from $B$, then we say the events $A$ and $B$ are dependent. In the special case where this probability stays the same (whether we select our outcome from at large or restrict our selection to just $B$) the events $A$ and $B$ are not dependent, or independent. In symbols:$$ P(A|B)=P(A)=P(A|\overline{B}) \Rightarrow A \text{ and } B \text{ are independent}$$while$$ P(A|B)< P(A)< P(A|\overline{B}) \Rightarrow A \text{ and } B \text{ are dependent.}$$
$\bf{Example:}$Suppose our sample space has 100 outcomes. 70 in $B$ (and so 30 in $\overline{B}$). Also suppose 50 outcomes are in $A$. If 35 outcomes are in $A \cap B$ (and so 15 are in $A \cap \overline{B}$), then $A$ and $B$ are independent since$$ P(A|B)=P(A)=P(A|\overline{B})=\frac{1}{2} $$If the 50 outcomes in $A$ are distributed between $B$ and $\overline{B}$ in any other way, $A$ and $B$ are dependent. For example, if 30 outcomes are in $A \cap B$ (and so 20 are in $A \cap \overline{B}$), then $A$ and $B$ are dependent since$$ \Big( P(A|B)=\frac{30}{70} \Big)< \Big( P(A)=\frac{50}{100} \Big) < \Big( P(A|\overline{B})=\frac{20}{30} \Big). $$(I.e. The probability of getting an outcome in $A$ "depends" on whether we choose from just $B$, at large, or just $\overline{B}$.)
$\endgroup$
