Show that if event A is completely independent of Event B

[TomJerry]

Problem : Show that if event A is completely independent of Event B and C then A is independent of BUC?

 

[dmatador]

Show some work. There are a few formulas about independence you can use to get you started.

 

[mathman]

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!

 

[mathman]

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!

 

[blue_raver22]

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.