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**,一枚硬币的结果es 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 t**wo 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**.