What is the difference between mutually exclusive and independent events?

Question

If events are mutually exclusive are they independent?

Answer 1

Definition of a mutually exclusive event



If event A happens, then event B cannot, or vice-versa. The two events "it rained on Tuesday" and "it did not rain on Tuesday" are mutually exclusive events. When calculating the probabilities for exclusive events you add the probabilities.



Independent events



The outcome of event A, has no effect on the outcome of event B. Such as "It rained on Tuesday" and "My chair broke at work". When calculating the probabilities for independent events you multiply the probabilities. You are effectively saying what is the chance of both events happening bearing in mind that the two were unrelated.



To be or not to be.....?



So, if A and B are mutually exclusive, they cannot be independent. If A and B are independent, they cannot be mutually exclusive. Simple isn't it? Or is it? This is where a lot of people go wrong in trying to work out probabilities as sometimes the status of two sets of probabilities are not as clear cut as it seems.



Taken directly from the source

Answer 2

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If A and B are mutually exclusive events then P(A U B) = P(A) + P(B) and P(A ∩ B) = 0, because the intersection of A and B is the empty set, i.e., A ∩ B = ∅. Two events are independent if: P(A | B) = P(A), this implies that P(A ∩ B) / P(B) = P(A) and thus P(A ∩ B) = P(A) * P(B) If A and B are mutually exclusive then P(A∩B) = 0 and P(A | B) = 0. This shows that mutually exclusive events A and B are not independent given P(A) > 0 and P(B) > 0. Examples: Roll two dice, let A be the event that both dice show a 1, 2, or 3. Let B be the event that the sum of the to dice is greater than 7. A and B are mutually exclusive because it is impossible for both events to happen at the same time. The are not independent because if A happens you know that B cannot happen. Knowing something about one of the events tells you about the other.

Answer 3

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RE:

What is the difference between mutually exclusive and independent events?

If events are mutually exclusive are they independent?

Answer 4

Independent Events Definition

Answer 5

mutually exclusive means two outcomes cannot happen simultaneously. For example, picking a random number - being odd and being even are mutually exclusive outcomes. Being even and a multiple of 3 are not mutually exclusive (i.e. 6). This is for ONE event with multiple possible outcomes.



Independent events means the outcome of one event has no impact on a second event. For example, the lottery numbers selected last Saturday have no impact on the lottery numbers selected this Saturday.

An example of dependent events would be picking two cards without replacing the first. The second card's probability is affected by whatever was chosen first since it wasn't replaced.



Determining mutual exclusivity and independence have nothing to do with each other.

Answer 6

Mutually exclusive events are events that cannot happen at the same time.



For example, the event "I roll a 1 on this six-sided die on my first roll" and "I roll a 2 on this six-sided die on my first roll" are mutually exclusive.



Independent events are events such that the outcome of one event does not affect the outcome of the second, and vice versa.



For example, the event "I roll a 1 on this six-sided die on my first roll" and "I roll a 2 on this six-sided die on my second roll" are independent.



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"Event 1 and Event 2 are Independent" can also be taken to mean that if you know whether or not Event 1 happened, then this doesn't change the probability that Event 2 happened.



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If two events are mutually exclusive, then if you know that Event 1 happened, then it does change the probability that Event 2 happened (to zero, because they can't have happened at the same time). So if two events are mutually exclusive, then they cannot be independent (except in certain trivial cases, such as both events being impossible).

Answer 7

every single event is always mutually exclusive whereas combined events can be independent or dependent, for example tossing a coin head and tail can not occur at the same time so they are mutually exclusive if two coins are tossed at the same time two events can be same but the probability of one event does not affect the probability of other event so it is independent.