probability – What is the difference between independent and mutually exclusive events?

The Question :

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Two events are mutually exclusive if they can’t both happen.

Independent events are events where knowledge of the probability of one doesn’t change the probability of the other.

Are these definitions correct? If possible, please give more than one example and counterexample.

The Question Comments :
  • They are, in a sense, completely opposite features. If $A$ and $B$ are independent, knowledge that $A$ occurred does not change the probabilities that $B$ may have occurred. Where as if $A$ and $B$ are disjoint, knowledge that $A$ occurred completely changes the probabilities that $B$ may have occurred by collapsing them to $0$.
  • I just noticed that the definitions in this question look like they’ve been taken from my answer here. (Not that I mind or anything.)
  • Consider taking out a card from a deck of $52$ playing cards. $S$: The card is a spade. $A$: The card is an ace. The two events are not mutually exclusive as there exists an Ace of Spades. $P(A) = \frac{4}{52}$ and $P(S) = \frac{1}{4}$. and $P(A\cap S) = \frac{1}{52} = \frac{4}{52} \frac{1}{4} = P(A) P(S)$
  • @alex.jordan If you wanted to highlight the analogy, you could say that independence and mutual exclusivity were defined by $P(AB)=P(A)P(B)$ and $P(A+B)=P(A)+P(B)$ respectively.
  • The book Counterexamples in Probability (Third Edition) by J. M. Stoyanov (Dover, 2013) is a treasure trove of information. In particular, Section 3 of Chapter 1 explores INDEPENDENCE OF RANDOM EVENTS.

The Answer 1

109 people think this answer is useful

Yes, that’s fine.

Events are mutually exclusive if the occurrence of one event excludes the occurrence of the other(s). Mutually exclusive events cannot happen at the same time. For example: when tossing a coin, the result can either be heads or tails but cannot be both.

$$\left.\begin{align}P(A\cap B) &= 0 \\ P(A\cup B) &= P(A)+P(B)\\ P(A\mid B)&=0 \\ P(A\mid \neg B) &= \frac{P(A)}{1-P(B)}\end{align}\right\}\text{ mutually exclusive }A,B$$

Events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of the other(s). For example: when tossing two coins, the result of one flip does not affect the result of the other.

$$\left.\begin{align}P(A\cap B) &= P(A)P(B) \\ P(A\cup B) &= P(A)+P(B)-P(A)P(B)\\ P(A\mid B)&=P(A) \\ P(A\mid \neg B) &= P(A)\end{align}\right\}\text{ independent }A,B$$

This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive. (Events of measure zero excepted.)

The Answer 2

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After reading the answers above I still could not understand clearly the difference between mutually exclusive AND independent events. I found a nice answer from Dr. Pete posted on math forum. So I attach it here so that op and many other confused guys like me could save some of their time.

If two events A and B are independent a real-life example is the following. Consider a fair coin and a fair
six-sided die. Let event A be obtaining heads, and event B be rolling
a 6. Then we can reasonably assume that events A and B are
independent, because the outcome of one does not affect the outcome of
the other. The probability that both A and B occur is

P(A and B) = P(A)P(B) = (1/2)(1/6) = 1/12.

An example of a mutually exclusive event is the following. Consider a
fair six-sided die as before, only in addition to the numbers 1
through 6 on each face, we have the property that the even-numbered
faces are colored red, and the odd-numbered faces are colored green.
Let event A be rolling a green face, and event B be rolling a 6. Then

P(B) = 1/6

P(A) = 1/2

as in our previous example. But it is obvious that events A and B
cannot simultaneously occur, since rolling a 6 means the face is red,
and rolling a green face means the number showing is odd. Therefore

P(A and B) = 0.

Therefore, we see that a mutually exclusive pair of nontrivial events
are also necessarily dependent events. This makes sense because if A
and B are mutually exclusive, then if A occurs, then B cannot also
occur; and vice versa. This stands in contrast to saying the outcome
of A does not affect the outcome of B, which is independence of
events.

The Answer 3

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Mutually exclusive event :- two events are mutually exclusive event when they cannot occur at the same time. e.g if we flip a coin it can only show a head OR a tail, not both.

Independent event :- the occurrence of one event does not affect the occurrence of the others
e.g if we flip a coin two times, the first time may show a head, but this does not guarantee that the next time when we flip the coin the outcome will also be heads. From this example we can see the first event does not affect the occurrence of the next event.

The Answer 4

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If I toss a coin twice, the result of the first toss and the second toss are independent.

However the event that you get two heads is mutually exclusive to the event that you get two tails.

Suppose two events have a non-zero chance of occurring.

Then if the two events are mutually exclusive, they can not be independent.

If two events are independent, they cannot be mutually exclusive.

The Answer 5

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Think simple,for independents events we have two events (two different events like tossing coin and rolling a disc,tossing two coins).So,probability of occurence of one does not effect the probability of occurence of other.In case of mutually exclusive events we have also two evevts(may be more than two) but difference is that the events are derived from the same events (rolling dices with even number red coloured and odd number green coloured.here both events have from the only single dice not for two).

The Answer 6

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Answers have been useful. You have learnt it.

I am just sharing my learning here, just because the discussion help us solidify our understanding. Usually we talk about coins, dies in terms of trying to either understand or explain probabilistic events. If I don’t use certain concepts for a while, I forget them. Then I hit another subject, module say biology where Mendelian inheritance (genetics) actually incorporate probability, specifically independent events and mutually exclusive events.

It was quite interesting and let me revise my own concept gaps. I hope this will let you think from another application point of view.

Product rule applies to independent events. You can be tall regardless of your skin colour. You can be fair regardless of your height. So you can be tall and fair at the same time, (no offence to anyone, just taking the dominant traits here) These are two independent events, that can come about together.

Another way, you like to swim, your brother/sister likes to play rugby. Both of you could be Olympians, racing in your competitions at the same time and taking two medals away together. You can’t swim in a rugby ground, so you will not influence his medal in anyway, so is he.

You like to play the piano, your friend likes to play the violin. Both of you can play together and conduct a concert or an orchestra (if you may). You don’t influence each other in anyway.

Thus, swim AND rugby taking place = p(swim) * p(rugby)

Sum rule applies to mutually exclusive events. Imagine, only one swimmer will be chosen from the country to represent in Olympics for 100m freestyle. Lets say you are from Australia, and coaches must select one person for the team. One person, either you or Kyle Chalmers.

In this case if you are selected, Kyle can’t and vice versa. Both of you can’t swim at the same time.

Thus probability of you OR Kyle swimming in Olympics = p(you) + p(Kyle)

Coming back to the genetics simple example for mutual exclusiveness, you can either be tall or short, fair or dark.

Maths is life, explains physics, chemistry, biology, and what not. So Good luck!

The Answer 7

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This question already has very good answers, I’m gonna add a visualization for independents events using some special diagrams. In these diagrams proportion of events to sample space represents their probability. Our sample space is a rectangle of 9x5 = 45 units:

enter image description here

We have event A (3x5) so P(A) = 3x5/9x5 = 15/45 = 1/3:

enter image description here

And event B (9x3) so P(B) = 9x3/9x5 = 27/45 = 3/5:

enter image description here

These two events intersect as:

enter image description here

𝐴∩𝐵 occupies 3x3 units:

enter image description here

𝑃(𝐴∣𝐵) = 𝑃(𝐴∩𝐵) / 𝑃(𝐵) so 𝑃(𝐴∣𝐵) = 9/27 = 1/3. But this is same as P(A)!

enter image description here

and 𝑃(𝐵∣𝐴) = 𝑃(𝐴∩𝐵) / 𝑃(𝐴) so 𝑃(𝐵∣𝐴) = 9/15 = 3/5. But this is same as P(𝐵)!

enter image description here

As in the two last diagrams, occurrence of one event doesn’t affect the probability of the other event, these two events are called independent. So knowledge about occurrence of one of them doesn’t affect our knowledge about probability of the other one. But this is not because they have nothing in common, on the contrary they are kinda in harmony by wiping out (the given event reduces sample space to itself, so it wipes out its complement) sample space in such a way that the other event proportion to given event doesn’t change. I’d like to remember them as perpendicular events.

The Answer 8

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It will be easier if we distinguish “mutually exclusiveness” from “independency” by considering the sample space in mind.

  • Two events that are compared for mutually exclusiveness must be from a single sample space. For example,

    • Tossing a coin twice. $A=\{HH\}$ is an event in which the head shows up twice and $B=\{TT\}$ is an event in which the tail shows up twice. Their share the same sample space $S=\{HH,HT,TH,TT\}$. As $A\cap B=\{\}$, they are mutually exclusive.
  • Two events that are compared for independency must be from two sample spaces. For example,

    • Tossing a coin twice. $A=\{H\}$ is an event in which the head shows up in the first throw and $B=\{T\}$ is an event in which the tail shows up in the second throw. The sample space for the first trial is $S_1=\{H,T\}$ and the sample space for the second trial is $S_2=\{H,T\}$ As $S_1=S_2$, they are independent.

The next additional questions are “is it possible to have 2 events that are”

  • “both mutually exclusive and independent?”

  • “both mutually exclusive and dependent?”

I will update this answer in the future to answer the additional questions above.

The Answer 9

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Events are Independent when happening of one does not influence happening of other. Eruption of volcano on Earth and orbit of Mars do not influence each other, so are independent events.

Growth of human population and preservation of many other species are mutually exclusive, as the one can only happen if the other does not happen.

Strictly speaking, mutually exclusive does not imply that one of them must happen. If there is a large asteroid impact on Earth, then neither human population grows nor endangered species are preserved.

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