Intersection of unindependent events

In summary, the equation -0.25 <= P(X \cap Y) - P(X)P(Y) <= 0.25 can be used to show the relationship between the joint probability of two events and the individual probabilities of those events, using Bayes' theorem. If the events are independent, the result will be 0, but if they are dependent or anti-dependent, the result will fall within the range of -0.25 to 0.25. Further exploration is needed to determine the exact relationship between the events in these cases.
  • #1
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Homework Statement



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-0.25 <= P( X [tex]\cap[/tex] Y ) - P( X )P( Y ) <= 0.25

for any events X, Y


Homework Equations


P( X [tex]\cap[/tex] Y ) = P( X )P( Y | X )
Bayes' theorem
Anything I missed?


The Attempt at a Solution



Obviously if X and Y are independent
P( X [tex]\cap[/tex] Y ) = P( X )P( Y )
so
P( X [tex]\cap[/tex] Y ) - P( X )P( Y ) = 0

but if they are not then I hit a wall. I've done pages of math but I go round in circles. I think there's some trick but I can't figure it out. Can anyone tell me what I'm missing?
 
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  • #2
If the events are totally independent, as you surmised, the result is 0.

What if, they are totally dependent, that is, P(X)=P(Y)?
Or if they are totally anti-dependent, that is, P(X)=1-P(Y)?
 

1. What is the intersection of unindependent events?

The intersection of unindependent events refers to the outcome or occurrence of two or more events happening at the same time or overlapping with each other. These events are considered to be unindependent because the outcome of one event does not affect the outcome of the other event.

2. How is the intersection of unindependent events calculated?

The intersection of unindependent events can be calculated by multiplying the probabilities of each event occurring. For example, if event A has a probability of 0.5 and event B has a probability of 0.3, the probability of both events occurring together is 0.5 x 0.3 = 0.15. This is because the two events are considered to be unindependent, so the probabilities can be multiplied together.

3. What is the difference between independent and unindependent events?

Independent events are events where the outcome of one event does not affect the outcome of another event. In contrast, unindependent events are events where the outcome of one event can affect the outcome of another event. For example, drawing two cards from a deck without replacement is an example of unindependent events, as the first card drawn affects the probability of the second card being drawn.

4. How can the intersection of unindependent events be used in real-life scenarios?

The intersection of unindependent events can be used to calculate the probability of two or more events happening together. This can be useful in various industries, such as insurance, where the probability of multiple events occurring at the same time needs to be taken into account to determine risk and coverage.

5. Can the intersection of unindependent events be greater than 1?

No, the intersection of unindependent events cannot be greater than 1. This is because the probability of an event occurring cannot be greater than 1 (100%). Therefore, the probability of two or more events occurring together cannot be greater than the probability of any single event occurring.

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