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.
asked Sep 22, 2014 at 5:24 1,513 4 4 gold badges 11 11 silver badges 12 12 bronze badges$\begingroup$ 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 $0$. $\endgroup$
Commented Feb 2, 2016 at 22:56$\begingroup$ 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.) $\endgroup$
Commented Feb 19, 2016 at 9:50$\begingroup$ 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. $P(A) = \frac<4>$ and $P(S) = \frac<4>$. and $P(A\cap S) = \frac = \frac<4> \frac <4>= P(A) P(S)$ $\endgroup$
Commented Mar 19, 2017 at 13:21$\begingroup$ @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. $\endgroup$
Commented Apr 11, 2017 at 18:21$\begingroup$ 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. $\endgroup$
Commented Nov 28, 2017 at 18:07Events 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.
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.
$$\left.\beginP(A\cap B) &= P(A)P(B) \\ P(A\cup B) &= P(A)+P(B)-P(A)P(B)\\ P(A\mid B)&=P(A) \\ P(A\mid \neg B) &= P(A)\end\right\>\text< independent >A,B$$
This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive. (Events of measure zero excepted.)
answered Sep 22, 2014 at 5:46 Graham Kemp Graham Kemp 131k 7 7 gold badges 54 54 silver badges 126 126 bronze badges $\begingroup$ "This of course means. " Events of probability zero excluded. $\endgroup$ Commented Sep 22, 2014 at 6:21$\begingroup$ Is there any connection between independent events and mutually exclusive events? I meant to ask "If $A$ and $B$ are mutually exclusive, what can be commented on the independence of $A$ and $B$ or vice versa." Or is there no such connection at all? I guess there is none. But just want to confirm. $\endgroup$
Commented May 23, 2015 at 15:22$\begingroup$ @Mahesha999 If two events are mutually exclusive, then they are NOT independent. $\endgroup$
Commented Jun 24, 2015 at 1:11$\begingroup$ @Anwar "Yes, that's fine" is the wrong answer to your question because it is not correct to say the "knowledge of the probability" changes the probability that an event happens. The "knowledge of the probability" never changes anything. The occurence of an event changes the probability as Graham explains in his answer. $\endgroup$
Commented Mar 12, 2017 at 7:59 $\begingroup$ If $A$ is independent from $B$, then it is independent from $\neg B$. $\endgroup$ Commented Oct 27, 2017 at 2:50 $\begingroup$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.