banner

Can two events with nonzero probabilities be both independent and mutually exclusive?

The question aims to answer if two events can be both independent and mutually exclusive simultaneously with non-zero probabilities. When we toss two coins, the result of one coin does not affect the other. if one outcome is head/tail, this doesn’t affect the result of another event. This means mutually exclusive events are not independent.

Expert Answer

No, two events cannot be independent and mutually exclusive at the same time.

The two events are mutually exclusive if they cannot occur at the same time. If the occurrence of one event does not affect the occurrence of the other event, the two events are independent. Therefore, two events cannot occur at the same time. This is because if one event occurs, the other event does not occur, so the second event is affected by the occurrence of the first event.

Let’s suppose $A$ and $B$ be two events. If these events are mutually exclusive, both cannot occur at the same time. The probability of both occurring at the same time is zero.

\[P(A\cap B)=0\]

If these two events are independent of each other, the probability that one of these events will occur is independent of whether the other event occurs. The probability that both will occur at the same time is the product of the probabilities of each occurrence.

\[P (A\cap B) = P (A) P (B)\]

How to get $P (A)P (B)$ equal to zero is if either $P(A)$ or $P(B)$ equals zero.

In that case, the events can be considered independent at the same time and mutually exclusive. To do this, disable one or both events if allowed.

Numerical Result

No, two events cannot be independent and mutually exclusive at the same time.

Example

Two independent events cannot be mutually exclusive unless the probability of one or both events is zero (that is, one or both events are not possible). Note that the occurrence of $A$ affects the occurrence of $B$ if the two events $A$ and $B$ are mutually exclusive.

More precisely: If $A$ occurs, $B$ does not occur. If $B$ occurs, $A$ does not occur. Therefore, the two mutually exclusive events are not independent.

Note: If the two events $A$ and $B$ are both independent and mutually exclusive, then the following equation is obtained:

\[P(A\cap B)=P(A)P(B) [Because\: A\: and\: B\: are\: independent\: events]\]

\[P(A\cap B)=0 [Because\: A\:and\: B\: are\: mutually\: exclusive\: events]\]

Combining these two equations gives us:

\[P(A)P(B)=0\]

This means that the probability of $P (A) = 0$, $P (B) = 0$, or both should be zero to make both events happen simultaneously.

Hence, two events cannot be both independent and mutually exclusive simultaneously with non-zero probabilities.

Previous Question < > Next Question

5/5 - (18 votes)