Mutually Exclusive vs Independent Events: Difference and Comparison

Events occur as a result of experiments that are termed random or uneven.

During a course of an experiment, the events are always kept tabs on by the mathematical function of probability.

In an experiment, many events have their probabilities measured, such as mutually exclusive, independent, dependent, simple, or compound.

Key Takeaways

1. Mutually exclusive events cannot coincide, while independent events do not affect each other’s probabilities.
2. In mutually exclusive events, the occurrence of one event means the other event cannot happen; independent events can both happen simultaneously.
3. The probability of both mutually exclusive events occurring is 0, while the probability of both independent events occurring is the product of their probabilities.

Mutually Exclusive vs Independent Events

Mutually exclusive events cannot occur at the same time, which means that if one event occurs, the other event cannot occur. Independent occurrences are those in which the occurrence of one event has no effect on the likelihood of the occurrence of the other.

Mutually exclusive, as the name suggests, gives an event type where the occurring event can’t be more than one at a given possible moment.

This means that the events that happen are all individual and unique at all times, and no recurring ones could be expected.

Given a particular time limit and within that, there can’t be more than a single experimental occurring, giving rise to a mutually exclusive event.

Independent events are what people normally mean when they refer to any event.

In this type of probability, more than one or even more than any number of events can take place without affecting another event that might have been taking place at the same time as the one in reference.

There are no limits to the number of occurrences that might take place together within a single experimental event.

Comparison Table

Parameters of Comparison	Mutually Exclusive Events	Independent Events
Do One Event Influence Another in the Same Environment?	Yes	No
Formula	P(A and B) = 0	P(A and B) = P(A) P(B)
Nature of Venn Diagram	The circles don’t overlap	Circles overlap
Simultaneous Occurrences	No	Yes
Other Names	Many such as disjoint events, etc	Not much

What is Mutually Exclusive Event?

Mutually exclusive events are often called disjoint events.

It always means an individual occurrence that isn’t accompanied by any other happening at the same time.

An event happening during a selected period has no chance to influence another happening along with it.

This is because such an event is always a single one. No two events are happening together.

But that event, for certain, influences the experimental surrounding around itself.

This technically means that there is no experimental occurrence happening simultaneously.

It defies some laws that people might see as general common sense.

In certain scenarios, an mutually exclusive event might seem impossible as those events need to be happening together simultaneously.

It is rare to have an event be categorized under the probability of being mutually exclusive.

The most common example of such an event is coin tossing.

During a single toss, the toss is likely to turn out to be either head or tail.

Never can a single toss result in both heads or tails. Of course, the coin can always land vertically without falling on one side.

But such cases are rare, and those events are categorized to a different probability factor.

This clearly shows that the occurrence of an individual event makes the happening of another event in the same period impossible.

In mutually exclusive events, all happenings are unique and have control over themselves.

It can’t exert a controlling element over another event.

What is Independent Event?

As the name suggests, an individual event holds no responsibility for the pattern of another event happening around it.

Two or more experimental occurrences can happen together, but they don’t affect each other in an independent event.

This probability is the most commonly seen event type around us, as most environmental occurrences happen irrespective of another.

Independent events don’t influence their surrounding to change to accommodate the event.

Neither does an independent hold any influence over other events taking place in its same environment.

This influence would be impossible to happen as all events in the probability of an independent event are naturally separated from each other.

The easiest example of an independent event is two coins tossed simultaneously beside each other.

The probability of heads and tails each is two, whereas the probability of one each is also the very same.

This clearly shows that one coin toss doesn’t hinder the probability of the coin toss happening at the same time beside it.

Any event happening independently has the upper hand to let all other events happening around it independently too.

This added advantage is also why most probability factors around us are also independent.

If in a sack filled with colored balls and two people pick up a ball each, either can pick the same or different colors.

This is all a magnificent mathematical probability showing the relative effects of events.

Main Differences Between Mutually Exclusive and Independent Events

1. While mutually exclusive events influence the occurrence of any other event if it takes place in the same environment, independent events don’t have such an influence.
2. Independent events can happen simultaneously, whereas mutually exclusive events can’t happen simultaneously.
3. In the Venn diagram, the circles overlap for independent events, while for mutually exclusive events, they don’t.
4. While the mathematical formula for mutually exclusive events equates to zero, that if independent events don’t and are always in a probability of two events.
5. Mutually exclusive events don’t occur at once for while independent events do.

References

1. https://www.stat.auckland.ac.nz/~iase/publications/icots2/Kelly.Zwiers.pdf
2. https://www.tandfonline.com/doi/abs/10.1080/00031305.1987.10475443