Show that if event A is completely independent of Event B

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In summary, if event A is completely independent of events B and C, it does not necessarily mean that A is independent of the union of B and C. This can be shown through counterexamples involving probability spaces and elementary events.
  • #1
TomJerry
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Problem : Show that if event A is completely independent of Event B and C then A is independent of BUC?
 
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  • #2
Show some work. There are a few formulas about independence you can use to get you started.
 
  • #3
TomJerry said:
Problem : Show that if event A is completely independent of Event B and C then A is independent of BUC?
Not true!

Counter example: Probability space is unit square, with probability = area.
A: 0≤x≤1/2, 0≤y≤1
B: 0≤x≤1, 0≤y≤1/2
C: two pieces C1 + C2
C1: 0≤x≤1/2, 0≤y≤1/2
C2: 1/2<x≤1, 1/2≤y≤1
P(A)=P(B)=P(C)= 1/2
P(AandB)=1/4, P(AandC)=1/4, A independent of B and C
P(BUC)=3/4, P(AandBUC)=1/4, while P(A)P(BUC)=3/8, not independent!
 
  • #4
Simpler counter example (actually a simplification of the above).
Consider a probability space with 4 elementary events k,l,m,n where each event has probability 1/4.
Let A={k,l}, B={k,m}, C={k,n}. Then:
P(A)=P(B)=P(C)= 1/2
A∩B={k}, A∩C={k}, A∩(BUC)={k}, while BUC={k,m,n}.
P(A∩B)=1/4, P(A∩C)=1/4, A independent of B and C
P(BUC)=3/4, P(A∩(BUC))=1/4 ≠ P(A)P(BUC)=3/8, not independent!
 
  • #5


To show that event A is completely independent of event B and C, we need to prove that the occurrence of event A does not affect the probability of event B and C happening. In other words, the probability of event A occurring is not influenced by the occurrence or non-occurrence of event B and C.

Now, to show that event A is independent of BUC, we need to prove that the occurrence of event A does not affect the probability of the combination of events B and C happening. In other words, the probability of event A occurring is not influenced by the combination of events B and C occurring or not occurring.

Since we have already established that event A is completely independent of event B and C, it follows that the probability of event A occurring is not affected by the occurrence or non-occurrence of event B and C. Therefore, the probability of event A occurring is also not affected by the combination of events B and C occurring or not occurring. This proves that event A is independent of BUC.
 

1. What does it mean for two events to be completely independent?

Two events are considered completely independent if the occurrence or non-occurrence of one event does not have any influence on the occurrence or non-occurrence of the other event. In other words, the probability of one event happening does not change based on whether the other event happens or not.

2. How is independence between two events mathematically represented?

The mathematical representation of independence between two events A and B is P(A|B) = P(A), where P(A|B) represents the conditional probability of event A given event B and P(A) represents the probability of event A occurring.

3. What is the significance of two events being completely independent?

The significance of two events being completely independent is that their relationship does not affect the outcome of each other. This allows for simpler calculations and predictions in statistical analysis and probability.

4. Can two events be both independent and dependent at the same time?

No, two events cannot be both independent and dependent at the same time. Independence and dependence are mutually exclusive concepts, meaning that if two events are independent, they cannot be dependent and vice versa.

5. How can we determine if two events are completely independent?

To determine if two events are completely independent, we can use the mathematical representation mentioned earlier (P(A|B) = P(A)) to calculate the conditional probability of one event given the other. If the result is equal to the probability of the first event occurring, then the events are completely independent.

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