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I would like to understand the difference between mutually exclusive events and independent events for the following example: 

    Two buckets A and B have the following:

    Bucket A: 3 black balls and 5 white balls
    Bucket B: 4 black balls and 2 white balls

One ball is drawn from each bag. Find the probability of getting 2 white balls.

Here are my questions:

  1. If E1 is an event for Bucket A and E2 an event for Bucket B. Why are these events "independent" and not "mutually exclusive"?
  2. What changes would you make to the problem to make the two events mutually exclusive?

Thanks for your help.

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  • $\begingroup$ Thanks, I "know" the definition of mutually exclusive and independent events. However, I just get tripped on questions like the one above. I need to understand why I get tripped. In other words, what is the fallacy in my argument. This will help me solve general probability problems and not the specific ones. Yes, I tagged as self-study. $\endgroup$
    – Rohit
    Commented Jul 9, 2015 at 0:00
  • $\begingroup$ I thought it to be mutually exclusive because within event E1 you cannot get a white and a black ball. The same applies to E2. Hence, I would think E1 and E2 are mutually exclusive. But they are not. In fact they are independent. So why am I wrong? Hope this clears my question. $\endgroup$
    – Rohit
    Commented Jul 9, 2015 at 0:02
  • $\begingroup$ The two events being considered are not "a white is drawn from bucket 1" and "a black is drawn from bucket 1", so the fact that you cannot get a white and a black ball from bucket 1 is irrelevant to the question about the second event involving a different bucket. Define a specific event for E1 and a specific event for E2 and consider the definition of mutually exclusive I gave -- i.e. ask yourself "can they both happen?" ... The examples in my answer were deliberately chosen to parallel the situation in your question, and if you read my answer with care, I do address your underlying issue. $\endgroup$
    – Glen_b
    Commented Jul 9, 2015 at 0:05
  • $\begingroup$ e.g. If E1 was "draw a black ball from bucket 1" and E2 was "draw a white ball from bucket 2" and in your experiment you drew 1 ball from each bucket ... could both E1 and E2 occur? $\endgroup$
    – Glen_b
    Commented Jul 9, 2015 at 0:11
  • $\begingroup$ Ah, I see. I was looking at mutually exclusive within E1 rather than between E1 and E2. So that was the mistake. So to answer the question E1 and E2 can both occur at the same time. Hence, they are not mutually exclusive. $\endgroup$
    – Rohit
    Commented Jul 9, 2015 at 1:03

1 Answer 1

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Mutually exclusive events can't both happen at the same time. If one happens, the other can't. e.g. If I roll a die and get an even number, I can't have got '3' with the same roll - the events "got an even number on that roll" and "got a 3 on that roll" are mutually exclusive.

When assessing whether two events are mutually exclusive, the basic question is to ask yourself "can they both happen?

Independent events are not affected by or related to each other (so that P(A|B)=P(A) and P(B|A)=P(B)). Consider I roll a die twice, and look at the events "got an even number on the first roll" and "got a 3 on the second roll"; now they can both happen, and it doesn't matter what happened for one event when figuring out the probability for the other -- the two are independent.

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  • $\begingroup$ Independence is defined as $P(A \cap B) = P(A) P(B)$, which is more general than the definition you give. $\endgroup$
    – dsaxton
    Commented Jul 9, 2015 at 0:49
  • $\begingroup$ @dsaxton Yes, I know this quite well, but for the purposes of my discussion (motivating a basic understanding of the distinction from mutual exclusivity), it's the conditional probability that is more directly helpful. The situations not covered by $P(A|B)P(B)$ will be a distraction from conveying a basic conceptual distinction. I was quite sure someone would come along to complain about this (and it's fine to do so), but I chose to do it this way quite deliberately. $\endgroup$
    – Glen_b
    Commented Jul 9, 2015 at 1:32

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