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Independent event and Mutually exclusive event are a pair of probability related terms. These terms are used to describe the existence of two events in a mutually exclusive manner. However, the definition of independent and mutually exclusive events is unclear, and there is no universally accepted definition of this term. Estimating probability and random variables using these terms is also not easy, and a lot depends on the definition of the independent event or mutually exclusive event. From the mathematical point of view, both these terms are used to describe several situations. These terms are very confusing in real life.
Event:
An event is a subset of a sample space or a set of possible outcomes. The events are treated as if they are unrelated to each other. A simple event is an event that can be represented by a single point in the sample space. For example, if there are four outcomes in a sample space, then the outcomes of 1,2 constitute a simple event.
Independent Event:
Independent events are defined as those events that certain conditions or another event cannot interlink.
Examples of independent events are listed below:
(i) The number of heads a coin will turn after flipping is non-interlinked with the initial coin value.
(ii) The problem of counting the number of persons in the stadium is not interlinked with the seating arrangement in the stadium.
(iii) The total result in an election is not interlinked with the candidate’s vote percentage, etc.
Thus, independent events are defined as those events whose probability can be obtained by the multiplication of the probability of each event. This rule is called the multiplication rule of probability, and it is also considered the most common definition.
An Independent event is an event like a coin toss in which the event is independent of the subsequent events. The occurrence of one event does not interfere with the probability of the success of other events. Consider two different events A and B are independent if and only if,
P(A) = P(A) = P(B) = P(B)= P (AB) = 0.5 (or ½).
Example: Suppose we flip a coin. If the coin turns up heads, it can be inferred that the next six flips are likely to be headed. Here, Heads and tails are the independent events. In this case, P (H) =½ and P (T) = ½.
Mutually Exclusive Event:
There are certain sets of events/experiments whose combinations cannot occur. These sets of events are known as mutually exclusive events. Mutually exclusive events are sometimes confused with independent events.
Mutually exclusive events examples are:
(i) The probability of a person being rich or poor is zero.
(ii) The probability of a man being either a bachelor or widower is 0. Events ‘bachelor’ and ‘widower’ are mutually exclusive since there cannot be any men who can be both ‘bachelor’ and ‘widower’.
(iii) The probability of a person belonging to one caste or belonging to another caste is 0.
Thus, mutually exclusive events are those whose probability cannot be obtained by multiplication and can only be obtained by the factorization rule. This is called the factorization rule of probability.
Mutually exclusive events are mutually exclusive or incompatible events, which means if one of the two events occurs, then all other possibilities become impossible.
Consider two different events, A and B. Mutually exclusive event is an event that can be true only if the first event is true or the second event is true. If A and B are mutually exclusive, then,
P(A) = P(A and B ) = P(B) = 0
Mutually exclusive events Examples: Suppose we take several students to a classroom. According to students’ responses, there are always two students who talk at high speed in every class. Two students always talk at high speed in the same class. We assume that two students talking at high speed are a mutually exclusive event.
Key differences between an independent event and mutually a mutually exclusive event:
- If Two events, A and B, are mutually exclusive, then they cannot happen together.
- Two events A and B, are independent if they are not related.
- Mutually exclusive events are independent of each other, just like the independent events.
- Independent events can happen simultaneously, while mutually exclusive events cannot happen together.
- Independent events are generally easier to understand than mutually exclusive events.
- If two mutually exclusive events occur, it may not happen that the two independent events occur simultaneously.
- Independent events can be easily related to each other, while mutually exclusive events are not related.
Conclusion:
A mutually exclusive event is one for which both occurrences can’t occur with equal probability. So, when there are two mutually exclusive events, and the probability of occurrence of each of them is known, then we can work on the conditional probabilities to find out the joint probability of the two events. Mutually exclusive events are independent of a set of mutually exclusive events. Events are called mutually exclusive or incompatible if they cannot occur simultaneously and are called independent if they cannot be said to be dependent on each other.
