Probability of the Union of Indepedent Events

P(AC)=1-P(A).In summary, when A1, A2, ..., An are independent events, the probability of their union is equal to 1 minus the product of their complements. This can be proven by applying De Morgan's Law and using the fact that the probability of the complement of an event is equal to 1 minus the probability of the event. This holds true for any number of events, not just two.
  • #1
Obraz35
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Homework Statement


Show that if A1, A2, ..., An are independent events then
P(A1 U A2 U ... An) = 1 - [1-P(A1)][1-P(A2)]...[1-P(An)]


Homework Equations


If A and B are independent then the probability of their intersection is P(A)P(B).
The same can also be said of AC and B.

The Attempt at a Solution


I have managed to prove this algebraically for the case where n=2. But I am having trouble trying to do it for the general case since it is not as easy to break down the union of A1 through An into a union of disjoint sets as it was for n=2.
 
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  • #2
Apply De Morgan's Law (for n events)
 

FAQ: Probability of the Union of Indepedent Events

1. What is the probability of the union of independent events?

The probability of the union of independent events is the sum of the individual probabilities of each event occurring. This means that we add the probabilities of each event and then subtract the probability of both events occurring together, since this would result in double counting.

2. How is the probability of the union of independent events calculated?

The probability of the union of independent events is calculated using the formula P(A or B) = P(A) + P(B) - P(A and B), where A and B represent two independent events.

3. What does it mean for events to be independent?

Two events are considered independent if the outcome of one event does not affect the outcome of the other event. This means that the probability of one event occurring does not change based on whether or not the other event occurs.

4. Can the probability of the union of independent events be greater than 1?

No, the probability of any event cannot be greater than 1. This means that the sum of the individual probabilities of the events and the probability of both events occurring together cannot be greater than 1.

5. How is the probability of the union of more than two independent events calculated?

The probability of the union of more than two independent events can be calculated using the formula P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C). This formula follows the same logic as the formula for two events, but includes all possible combinations of the events occurring together.

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