Proving Independence: Probability Theory of A and B in a Simple Proof"

In summary, the students are discussing proving the independence of \bar A and \bar B when A and B are also independent. They are looking for guidance and the professor later posts the solution.
  • #1
FrogPad
810
0
If A and B are independent, prove that [itex] \bar A [/itex], [itex] \bar B [/itex] are independent.

Could someone help me start this. It's due tomorrow. I managed to prove that [itex] \bar A [/tex] is independent with [itex] B [/itex] and that [itex] \bar B [/itex] is independent with [itex] A [/itex], but I can't get the last one (the question I put above). Just a little nudge would be good. I've been going in circles trying stuff, from De Morgans law, to every identity I can think of.

Thanks
 
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  • #2
What are you using as the definition that A and B are independent?
 
  • #3
jing said:
What are you using as the definition that A and B are independent?
For A and B being independent,
P[AB]=P[A]P
 
  • #4
The professor posted the solution. Later :)
 

FAQ: Proving Independence: Probability Theory of A and B in a Simple Proof"

What is the significance of proving independence in probability theory?

In probability theory, independence between two events A and B means that the occurrence of one event does not affect the likelihood of the other event occurring. Proving independence allows us to make accurate predictions and calculations in situations where multiple events may be occurring simultaneously.

How is independence between two events A and B mathematically represented?

In probability theory, independence between two events A and B is mathematically represented as P(A ∩ B) = P(A) x P(B), where P(A ∩ B) represents the probability of both events A and B occurring, and P(A) and P(B) represent the individual probabilities of events A and B occurring.

What is a simple proof for proving independence between two events A and B?

A simple proof for proving independence between two events A and B involves using conditional probability. If P(A|B) = P(A), where P(A|B) represents the probability of event A occurring given that event B has occurred, then A and B are independent. This is because the occurrence of event B does not affect the likelihood of event A occurring.

Can independence be proven for more than two events?

Yes, independence can be proven for more than two events. The same principle of using conditional probability applies. If the probability of an event A occurring is not affected by the occurrence of events B, C, D, etc., then A is independent from those events.

What are some real-life examples of independent events?

Rolling a die and flipping a coin are two independent events. The outcome of rolling the die does not affect the outcome of flipping the coin. Another example is choosing a card from a deck and then choosing another card without replacing the first one. The first card chosen does not affect the probability of choosing the second card.

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