Hi I’m not sure if this is the place to ask, but I’m struggling to understand the difference between disjoint events and independent events. Does one imply the other or are they completely unrelated? Any clarification would be greatly appreciated!

*Two disjoint events* are necessarily mutually exclusive, [and](https://math.stackexchange.com/a/4419308/21813) *if they also have positive probabilities then they must be dependent.* – ryang Apr 03 '22 at 12:39
3 Answers
$A$ and $B$ are disjoint if $A\cap B=\varnothing $.
$A$ and $B$ are independent if $\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)$.
In particular, if $A$ and $B$ are disjoint and both has non zero probability to occur, then, they won't be independents. Finally, a nul set will be independent to all events.
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So am I right in assuming that two disjoint events are dependent because the occurrence of one depends on whether or not the other has occurred (as they can’t occur simultaneously)? – Ruby Pa Aug 30 '20 at 08:55

if $A$ and $B$ are disjoint and $\mathbb P(A),\mathbb P(B)>0$, then they are dependent. But if for example $\mathbb P(A)=0$ then $A$ is independent to all event. @RubyPa – Surb Aug 30 '20 at 09:05
Hint: Two independent events $A, B$, each with positive probability $p_A, p_B$ respectively, will have positive probability $p_A * p_B > 0$ of both happening. If $A, B$ are disjoint instead, what is the probability that they both happen?
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If $A$ and $B$ are independent, it means that the occurrence or not of one has no bearing on the occurrence or not of the other.
If $A$ and $B$ are disjoint, on the other hand, this is not the case.
Because if $A$ occurs, for example, then you know that $B$ does not occur. That makes the occurrence of $B$ dependent on the occurrence of $A$. And the other way about.
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