Why are mutually exclusive events not independent?

burgerandcheese

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Hi, from what I understand:

When two events A and B are mutually exclusive, then if one event occur, the other event cannot. Or if we draw a Venn diagram, A and B do not intersect.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = P(A) + P(B)

Events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of other(s). For example, when selecting 3 students out of a group of 5 students with replacement, then the probability of each trial is the same (it is possible to choose the same student multiple times).

P(A ∩ B) = P(A) * P(B)

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I've done a couple of searches online and I found that "mutually exclusive events are not independent, and independent events cannot be mutually exclusive". I'm frustrated because I still cannot understand why. I imagined it like this: if two events are independent, then they must intersect on the Venn diagram. But for mutually exclusive events, they do not touch. Therefore mutually exclusive events are not independent.
 
Events A and B are independent if A occurring has no influence on whether B occurs or not and vice-versa. But if A and B are "mutually exclusive" then A occurring prevents B from occurring.
 
But if A and B are "mutually exclusive" then A occurring prevents B from occurring.

So, since A occurring prevents B from occurring, that means A has an influence on B occurring? Is that why mutually exclusive events cannot be independent, because they influence the occurrence of another event?
 
So, since A occurring prevents B from occurring, that means A has an influence on B occurring? Is that why mutually exclusive events cannot be independent, because they influence the occurrence of another event?

I wouldn't say one event prevents the other from occurring. Seems too specific. Just think of it in terms of a 'connection' between events because of a particular setup of the experiment.
E.g. Experiment 1: flip 2 coins - first coin's outcome doesn't tell us anything about the second - all outcomes are independent.
Experiment 2: tape the coins together. By looking at the first coin we'll know exactly which side of the second coin is up. Now some outcomes are mutually exclusive.
Or: Event A is 'rainy day', event B is 'it's Friday' - independent events. Compare to: event A is 'rainy day', event B is 'no cloud in the sky' - mutually exclusive.
 
I've done a couple of searches online and I found that "mutually exclusive events are not independent, and independent events cannot be mutually exclusive". I'm frustrated because I still cannot understand why. I imagined it like this: if two events are independent, then they must intersect on the Venn diagram. But for mutually exclusive events, they do not touch. Therefore mutually exclusive events are not independent.

The claim you are asking about is just a little too strong.

Look at the definition of independence that you quoted:

P(A ∩ B) = P(A) * P(B)

If A and B were also mutually exclusive, this would become

0 = P(A) * P(B)

This can only happen if either A or B has probability zero. So there is a case in which events can be both mutually exclusive and independent: when one of them is impossible. That just isn't a very useful situation.
 
To build a bit on Dr. Peterson's post:

\(\displaystyle P(A) \ge 0 \text { and } P(B) = 0 \implies P(A \text { and } B) = 0 \implies P(A \text { and } B) = P(A) * P(B).\)

But that is not true if both probabilities are positive. In that case, the general rule is.

\(\displaystyle P(A) > 0 < P(B) \implies P(A|B) * P(B) = P(A \text { and } B) = P(B|A) * P(A).\)

Now we consider independent events.

\(\displaystyle A \text { and } B \text { are independent } \iff P(A|B) = P(A) \text { and } P(B|A) = P(B).\)

\(\displaystyle A \text { and } B \text { are independent and } \text P(A) > 0 < P(B) \implies P(A \text { and } B) = P(A) * P(B) > 0.\)

Now let's consider mutually exclusive events.

\(\displaystyle A \text { and } B \text { are mutually exclusive } \iff P(A|B) = 0 \text { and } P(B|A) = 0 \implies\)

\(\displaystyle P(A \text { and } B) = 0.\)

Now the probabilty of A and B cannot be zero and also greater than zero. So we have the following theorems:

If the probabilities of A and B are both positive and A and B are independent, the probability of A and B is greater than zero.

If A and B are mutually exclusive, the probability of A and B is zero.

Thus if P(A) > 0 < P(B), A and B are not both independent and mutually exclusive.

If all these seems abstract, consider this specific example. The probability of rolling a 1 on a single throw of a fair die is 1/6. The probability of rolling a 2 on a single throw of a fair die is 1/6. Nothing in arithmetic precludes us from multiplying those two fractions and getting 1/36. But do we ever get 1 and 2 on a single roll of a fair die? Obviously not. The two probabilities are mutually exclusive, and the probability of their joint occurrence is zero, not 1/36.

Clearer now?
 
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Hi, from what I understand:

When two events A and B are mutually exclusive, then if one event occur, the other event cannot. Or if we draw a Venn diagram, A and B do not intersect.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = P(A) + P(B)

Events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of other(s). For example, when selecting 3 students out of a group of 5 students with replacement, then the probability of each trial is the same (it is possible to choose the same student multiple times).

P(A ∩ B) = P(A) * P(B)

--

I've done a couple of searches online and I found that "mutually exclusive events are not independent, and independent events cannot be mutually exclusive". I'm frustrated because I still cannot understand why. I imagined it like this: if two events are independent, then they must intersect on the Venn diagram. But for mutually exclusive events, they do not touch. Therefore mutually exclusive events are not independent.
As you said: When two events A and B are mutually exclusive, then if one event occur, the other event cannot. Or if we draw a Venn diagram, A and B do not intersect.

Now two events are independent if the occurrence of one event does not influence the occurrence of other. Raining now in California does not influence whether or not it is raining now in New York, so raining now in California and raining now in New York are two independent events.

Now since these two independent events (raining in NY and raining in CA) can happen at the same time, then they are not mutually exclusive.


Now consider two mutually exclusive event: Raining outside your house right now and not raining outside your house right now. Clearly they both can't happen at the same time. But are they both independent events. The answer is no, since if you know that it is raining outside your home right now then you are sure that it is not the case that it is not raining outside your home right now. One influenced the other!

Now, can you have both independent and mutually exclusive at the same time. Great question. Let's look at the definitions. We need P(A ∩ B) = 0 and P(A ∩ B) = P(A) * P(B). That is P(A) * P(B) = 0. So one or both of P(A) & P(B) must be 0.
Think about what this will mean.
 
As you said: When two events A and B are mutually exclusive, then if one event occur, the other event cannot. Or if we draw a Venn diagram, A and B do not intersect.
Two of the biggest names in probability/statistics of the last century are Larsen& Marx. Here is their definition of mutually exclusive events: Events A & B defined on the same space are said to be mutually exclusive if they have no out comes in common.
Please note the careful wording of that definition: no out comes in common.
Consider the toss of two dice. The event \(\displaystyle A\) is the sum(total) is odd; the event \(\displaystyle B\) is that both faces show odd outcomes. Clearly event \(\displaystyle A\) can happen if and only if event \(\displaystyle B\) does not happen. So they are mutually exclusive. \(\displaystyle 2\cdot\left\| {\left\{ {1,3,5} \right\} \times \left\{ {2,4,6} \right\}} \right\| = 18\) so that \(\displaystyle \mathscr{P}(A)=0.5\)
So likewise \(\displaystyle \mathscr{P}(B)=0.25\). BUT
\(\displaystyle \mathscr{P}(A\cap B)=0\) & \(\displaystyle \mathscr{P}(A)\cdot\mathscr{P}(B)=0.125\)
 
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If events A and B are independent, it means that whether or not one occurs does not affect whether or not the other occurs. In other words, observing that A has occurred gives you zero additional knowledge as to whether B has occurred, and vice versa.

If events A and B are mutually-exclusive, it means that if one of them occurs, the other one cannot occur. So the events can't be independent, because the occurrence of one is highly-dependent on (strongly affects) the occurrence of the other one. If you observe that A has occurred, then you know that B cannot have occurred, and vice versa.

That's it. That's all there is to it. You can write down a bunch of fancy symbols involving set theory/operations if you want, but at the end of the day that's all the explanation needed. Two events clearly can't be independent if they influence each other's probability of happening (and more specifically, in this case, prevent or exclude each other from happening).
 
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HallsofIvy, lev888, Dr.Peterson, JeffM, Jomo, pka, j-astron

Thank you so much for replying. I understand now!
 
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