Mutually Exclusive and Independent Events

Independent, dependent, and mutually exclusive events all occur in Probability. It’s crucial to understand why these events differ from one another.

Probability and Events

In a Random Experiment, the probability is a measure of the possibility that an event will occur. Probability is expressed as a number between 0 and 1, with 0 denoting impossibility and 1 denoting certainty. The greater the chance of something happening, the more likely it is to happen.

A random experiment can have multiple outcomes, one of which is an event. It’s possible that events are connected in some way. Mutually exclusive and independent are two major ways in which events can be linked.

Mutually Exclusive and Independent Events

Let us see this example: when a coin is tossed in the air, there are two possible outcomes: either the coin will land on its head or the coin will land on its tail. As a result, in this scenario, both events are mutually exclusive. However, if two different coins and then flip them at the same time, the occurrence of a Head or Tail on both coins is independent of the tossing of the other.

• Mutually Exclusive Events

Mutually exclusive events occur when the occurrence of two events is not synchronous.

In the case of mutually exclusive events, the sets will not overlap.

Let’s take the toss of a coin as an example. When we toss a coin, let A represent the outcome if the coin lands on heads, and B represent the outcome if the coin lands on tails.

Events A and B are mutually exclusive in a single fair coin toss, meaning the outcome could be either tails or heads. We’re not going to be able to have both heads and tails in a simultaneous timeframe.

• Independent Events 

An independent event occurs when the occurrence of one event does not influence the occurrence of another.

In the case of independent events, the sets will overlap.

If the following statements are true, two events are independent:

P(A|B) = P(A)

P(B|A) = P(B)

P(A AND B) = P(A)P(B)

Two occurrences A and B are not dependent on each other if knowing that one occurred has no bearing on the likelihood that the other will occur. The outcomes of two roles of a fair die, for example, are separate events. The outcome of the first roll has no bearing on the probability of the second roll’s outcome. You must only exhibit one of the aforementioned conditions to demonstrate that two events are independent.

Difference between Mutually exclusive and independent events

We could think that the definitions of mutually exclusive events and independent events are the same at first. However, both are distinct events that occur in probability; let us examine the differences.

Mutually Exclusive Events

• Mutually exclusive events occur when the occurrence of two events is not synchronous.

• In the case of mutually exclusive events, the sets will not overlap.

• Mutually exclusive events are defined as P(A and B) = 0.

Independent Events

• An independent event occurs when the occurrence of one event does not influence the occurrence of another

• In the case of independent events, the sets will overlap.

• Independent events are defined as :  P(A and B) = P(A) P(B)

The distinction between mutually exclusive and independent events is illustrated in the following Venn diagram:

We can clearly see, In the case of mutually exclusive events, the sets will not overlap, In the case of independent events, the sets will overlap.

Conclusion

With the help of examples, we studied the terms Probability and Events, Mutually Exclusive and Independent Events, and the key differences between Mutually Exclusive and Independent Events in this article. When two events cannot occur at the same time, they are mutually exclusive, whereas when two events are independent, the occurrence of one does not affect the occurrence of the others.