The concepts of independence and exclusivity of events are intertwined in probability. Often, people get confused between these two concepts. So, the objective of this post is to clarify the difference between the two. Consider two events ${A}$ and ${B}$. The conditional probability of B when A occurs can be written as:
Now, let us say when the event $A$ occurs the event $B$ does not occur. For example: let $A$ be the event that it is day now and let $B$ be the event denoting that it is night now. So, can the events $A$ and $B$ ever occur together? The simple answer is no. This means that when event $A$ occurs, the probability of occurring of event $B$ is zero. As a result, both of these events never ever occur together. This means that both these events are exclusive. In simple terms, exclusivity of two events implies when one of the event occurs, the other event never occurs. So, in the first equation
But are these exclusive events independent? Before answering this question, let us first look at what independence of events means. Again, consider two events $C$ and $D$. If these two events are independent, occurrence of one event does not affect the occurrence of the other event. For example consider $C$ and $D$ to be the outcome of a coin toss. Let $C$ denote Heads when you throw the coin once while $D$ denote heads when you throw the coin again. So, you throw the coin once and you get a heads i.e. event $C$ occurs. Does occurrence of event $D$ in any way change? It does not. This means that occurrence of $C$ has in no way altered the chances of occurrence of event $D$. Mathematically, two events are said to be independent if:
Now, let us look at exclusive events and independence together. When the two events are exclusive, it means that one of the events (when it occurs, see the event $A$) alters the chances of occurrence of the other event (see the event $B$, which occurs with zero probability). Therefore, both the events are not independent. Thus, in general exclusivity implies events are not independent. There is another way of look at this problem. Since, we have stated that the events do not occur together, their intersection is zero (cf. second equation). Thus, in the first equation:
${ P \left( A=a|B=b \right) = \dfrac{ P \left( A = a \cap B = b \right) }{P \left( B = b \right)}}$
Now, let us say when the event $A$ occurs the event $B$ does not occur. For example: let $A$ be the event that it is day now and let $B$ be the event denoting that it is night now. So, can the events $A$ and $B$ ever occur together? The simple answer is no. This means that when event $A$ occurs, the probability of occurring of event $B$ is zero. As a result, both of these events never ever occur together. This means that both these events are exclusive. In simple terms, exclusivity of two events implies when one of the event occurs, the other event never occurs. So, in the first equation
${P \left( A = a \cap B = b \right) = 0}$
But are these exclusive events independent? Before answering this question, let us first look at what independence of events means. Again, consider two events $C$ and $D$. If these two events are independent, occurrence of one event does not affect the occurrence of the other event. For example consider $C$ and $D$ to be the outcome of a coin toss. Let $C$ denote Heads when you throw the coin once while $D$ denote heads when you throw the coin again. So, you throw the coin once and you get a heads i.e. event $C$ occurs. Does occurrence of event $D$ in any way change? It does not. This means that occurrence of $C$ has in no way altered the chances of occurrence of event $D$. Mathematically, two events are said to be independent if:
${ P \left( A=a|B=b \right) =P \left( A = a \right)}$
Now, let us look at exclusive events and independence together. When the two events are exclusive, it means that one of the events (when it occurs, see the event $A$) alters the chances of occurrence of the other event (see the event $B$, which occurs with zero probability). Therefore, both the events are not independent. Thus, in general exclusivity implies events are not independent. There is another way of look at this problem. Since, we have stated that the events do not occur together, their intersection is zero (cf. second equation). Thus, in the first equation:
${ P \left( A=a|B=b \right) = 0 \neq P \left( A=a \right) }$
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