Disjoint vs. Independent Events: What's the Difference?

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Discussion Overview

The discussion revolves around the differences between disjoint and independent events in probability theory, focusing on their definitions, implications for calculations, and examples. Participants explore how these concepts affect the calculation of probabilities, particularly in relation to unions and intersections of events.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants define disjoint events as mutually exclusive, indicating that if one event occurs, the other cannot, leading to P(A ∩ B) = 0.
  • Others explain that independent events do not affect each other's occurrence, expressed as P(A ∩ B) = P(A)P(B).
  • There is a question about the union of events, specifically whether the formula P(A ∪ B) = P(A) + P(B) applies to both disjoint and independent events.
  • Some participants assert that the union formula holds for disjoint events but not for independent events, which requires a different formula.
  • A participant raises a concern about the characterization of disjoint events as a strong form of dependence, suggesting it may be confusing.
  • Another participant argues that while coin tosses are independent, the events of heads and tails from a single toss are disjoint, leading to a discussion about the nature of dependence in this context.
  • There is a clarification that two events can only be both independent and disjoint if one has a probability of zero.
  • Some participants express uncertainty about the implications of independence and disjointness in the context of probability calculations.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of disjoint and independent events, with no consensus reached on certain points, particularly regarding the characterization of dependence and the conditions under which events can be both independent and disjoint.

Contextual Notes

Some discussions involve assumptions about the nature of events and their probabilities, particularly in the context of coin tosses and the interpretation of independence versus disjointness.

stukbv
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What is the difference between disjoint and independent events, how will the 2 affect calculations involving them?
 
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Disjoint events are mutually exclusive, which is a strong form of statistical dependence (so if you know event A occurred you know that B definitely did not occur and vice versa), meaning
P(A\cap B) = 0

For events to be independent on the other hand, knowing if one event occurred does not give you any information about if the other event occurred, which is generally expressed as
P(A\cap B) = P(A) P(B)
 
Ok, what about the union, i.e what aboue p(A U B ) if disjoint or independent, are we allowed to sum for both?
 
No, that only works for disjoint events, generally
P(A\cup B) = P(A) + P(B) - P(A\cap B)
so for disjoint events the third term is zero.
 
For independent events we do have the nice formula

P(A\cup B)=P(A)+P(B)-P(A)P(B)
 
sfs01 said:
Disjoint events are mutually exclusive, which is a strong form of statistical dependence (so if you know event A occurred you know that B definitely did not occur and vice versa), meaning
P(A\cap B) = 0
P(A\cap B) = P(A) P(B)

Well, tossing a fair coin leads to series of events that are both independent and disjoint. I wouldn't say that fair coin tosses are in any way dependent on each other.

In the usual sense, statistically independent events are not either/or outcomes. So for the tossing of a fair coin, the probability of H or T is exactly 1,and the third term is zero, not P= 1/4. However, as you say, if P(A) and P(B) are the probabilities of random independent events which are not mutually exclusive, then the sum of the probabilities is P(A)+P(B)-P(A)P(B); that is, the probability of A or B, less the probability of A and B.

I was just concerned that your description of disjoint events as representing a strong form of dependence might be confusing to some.
 
Last edited:
SW VandeCarr said:
Well, tossing a fair coin leads to series of events that are both independent and disjoint. I wouldn't say that fair coin tosses are in any way dependent on each other.

In the usual sense, statistically independent events are not either/or outcomes. So for the tossing of a fair coin, the probability of H or T is exactly 1,and the third term is zero, not P= 1/4. However, as you say, if P(A) and P(B) are the probabilities of random independent events which are not mutually exclusive, then the sum of the probabilities is P(A)+P(B)-P(A)P(B); that is, the probability of A or B, less the probability of A and B.

I was just concerned that your description of disjoint events as representing a strong form of dependence might be confusing to some.
Which events are meant to be both independent and disjoint in this case? The only way I can see that two events can ever be both independent and disjoint is if one of them has probability zero.

The main events for the coin tosses are
- the two events representing the results of each individual coin toss, which are disjoint, but not independent (since P(H \cap T) = 0 \ne P(H) P(T))
- the results of subsequent coin tosses, which are independent, but not disjoint, if they are modeled as part of the same sigma algebra (P(first toss is H) doesn't exclude P(second toss is H) or P(second toss is T))

Or did I miss any events you considered?

Of course, H and T don't have the kind of dependence you normally have with random variables, but can't see anything wrong with calling them dependent. All I meant by strong dependence (which maybe isn't a very well-defined concept as I used it) was that the probability of one event conditional on the other is zero for disjoint events, rather than just changing the original probability a bit as most types of dependence do.
 
Ahhh i seee, so if disjoint, they cannot both occur, so the intersection is the empty set
and if they are independent the probability of the intersection is equal to the product of their probabilities.

So when we have evens and we say the probability of their union is equal to their sum, does this mean they are independent or disjoint ?
 
stukbv said:
So when we have evens and we say the probability of their union is equal to their sum, does this mean they are independent or disjoint ?

Disjoint, but not independent in the statistical sense since they are mutually exclusive.
 
  • #10
What is the difference between independent and NONdisjoint events?
 

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