Mutually Exclusive and Independent Events

Probability is a mathematical concept that is a vital part of statistics and has become a full-fledged discipline. A random experiment in probability is an event that generates a certain outcome, it is purely based on chance, or we may call it luck. In an event, there are various types of Probabilities or events like simple, compound, exhaustive, independent, dependent, mutually exclusive, equally likely, etc. When an event cannot possibly occur simultaneously, they are called mutually exclusive.

On the other side, if an event is unaffected by other events, they are called independent events. Let’s know more about these mutually exclusive and independent events in the article below.

Mutually Exclusive Event

Mutually exclusive events cannot occur simultaneously, which means where the results of one event decide on the occurrence or non-occurrence of the other events. Such events cannot be performed simultaneously or determine the final results at the same time. Hence, the happening of one event determines the happening of another event. These are commonly referred to as disjoint events.

Let’s understand this with the help of an example of tossing a coin, where the obvious result would either be ahead or a tail. Unfortunately, both the head and tail cannot occur together at a time. Let’s take another example, suppose if a company wants to purchase a Machine, for which it has two options Machine A and B. The more profitable machine in terms of cost-effectively and productivity will be selected. The selection of machine A will automatically result in the rejection of machine B and vice versa here; the outcome is not my chance but biased to the profit.

Independent Event

As the name itself suggests, independent events are those events in which the probability of one event does not determine the probability of the occurrence of the other event. Every event is independent to occur. The occurrence or non-occurrence of such an event has no control over the occurrence or non-occurrence of another event. The simple product of their independent probabilities is equal to the probability that both events will occur.

For example, if you toss a coin twice, the tail is the outcome in the first chance and the tail in the second, the events are independent. Let’s see another example for this, if you roll a dice twice, 6 in the first chance and 3 in the second, the events are independent.

Basic Difference between Mutually Exclusive and Independent Events

The basic conceptual differences between the mutually exclusive and independent events are explained as under:

• Mutually exclusive events the events when their occurrence is not simultaneous. The result of one event determines the happening or non-happening of another event. When the occurrence of one event does not control the occurrence, such events are called independent events
• In mutually exclusive events, the occurrence of one event will result in the non-occurrence of the other. On the other hand, in independent events, the occurrence of one event will have no control over the occurrence of the other
• Mutually exclusive events are mathematically represented as P(A and B) = 0

Whereas independent events are mathematically represented as P (A and B) = P(A) P(B)

• Talking about the Venn diagram representation, the two or more sets do not overlap in the case of mutually exclusive events. In contrast, if we talk about independent events, the sets can overlap if the events are the same or repeated

Differences

Conclusion

Finally, we can conclude that when an event can not occur simultaneously, they are called mutually exclusive. On the other side, if an event is unaffected by other events, they are called independent events. These topics are briefly discussed in the article above. We also discussed the difference between the two above.