Disjoint vs. Independent Events in Probability

  • Thread starter Thread starter torquerotates
  • Start date Start date
  • Tags Tags
    Independence
AI Thread Summary
In probability, disjoint events are mutually exclusive, meaning if one event occurs, the other cannot, resulting in a probability of zero for the second event. In contrast, independent events imply that the occurrence of one event does not influence the probability of the other occurring. For independent events, the relationship is defined by the equation P(A ∩ B) = P(A)P(B). If two events are disjoint, then their intersection is empty, leading to the conclusion that they can only be independent if at least one of the events has a probability of zero. Understanding the distinction between disjoint and independent events is crucial for accurate probability calculations.
torquerotates
Messages
207
Reaction score
0
In probability is there a difference between sets that are disjoint and sets that are independent.
 
Physics news on Phys.org
I don't know what it means for sets to be independent! I think you are asking about events that are mutually exclusive or independent. Two events, as sets of possible outcomes, are disjoint if and only if they the events are mutually exclusive. but you can't use the word "independent" in that way.

Yes, there is a very large difference! If two events are independent that means knowing one happens doesn't affect the probability that the other happens. If two events are mutually exclusive know that one happens means the probability of the other is 0! That certainly affects their probability.

If I roll a single die, what is the probability it comes up "5"? If I tell you the result was an even number what does that tell you about the probability it was a "5"?
 
In the measure-theoretic axiomatization of probability, you can regards sets as events, so this question is somewhat well-formed. And yes, there is a difference. You would say events (sets) A and B are independent if P(A n B)=P(A)P(B). If A and B are disjoint, then A n B = emptyset, so P( A n B ) = 0, so A and B are independent iff P(A)=0 or P(B)=0.
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
View full post »

Similar threads

I Multicollinearity and Interactions
Replies
5
Views
2K
I Do These Probability Conditions Imply Independence of Events A, B, and C?
Replies
7
Views
2K
A A game of seemingly mutually independent events
Replies
7
Views
2K
I Axiom of Choice: Disjoint Family ##\Rightarrow ## Power Set
Replies
10
Views
2K
Disjoint vs. Independent Events: What's the Difference?
Replies
9
Views
20K
I Independent events and variables
Replies
12
Views
2K
MHB What is the Probability of a Tetrahedron Landing on a Specific Colored Side?
Replies
3
Views
1K
A Multiparticipant events and pairwise probabilities
Replies
16
Views
2K
B What is Conditional Probability and its Properties?
Replies
3
Views
2K
I How to show these random variables are independent?
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top