Independent Events Article byNiti Gupta For eg:
Source: Independent Events (wallstreetmojo.com)

Independent Events Definition

Independent event is a term widely used in statistics, which refers to the set of two events in which the occurrence of one of the events doesn’t impact the occurrence of another event of the set. In other words, these are those events that don’t provide any information about the occurrence or non-occurrence of other events.

Explanation

In a usual scenario, the occurrence or non-occurrence of a particular event may provide an insight into other events. However, the same is not the case in independent events, since the occurrence or non-occurrence of one event is not going to provide any idea or information about the existence of another event. Thus, the outcome of one of the events is not dependent on the outcome of another event in the same set.

Examples of Independent Events

The concept can be well understood with the help of a few examples –

• We take two coins and then toss them. The event of the appearance of tail or head on one coin is not decisive of the appearance of tail or head on another coin. Thus, tossing two coins simultaneously or tossing the same coin twice can be said to independent events. The reason is that the probability of each outcome (i.e., head or tails) is 50% each time and is not dependent on the last toss.
• Similarly, when we take two dices and roll them, the resultant number on one dice does not decide the resultant number on the second dice. As a result, the rolling of two dices is another example.

Rules

There is a multiplication rule in probability which can be tested upon to identify whether the two events are independent or not.

Multiplication rules state that, if two events are independent, then:

P(A|B) = P(A)

This mathematical connotation denotes that two events, named A and B, are said to be independent when the probability of event A, given that event B occurs, is equal to the probability of event A. Its because, in the case of independent events, the occurrence or non-occurrence of an event doesn’t decide the occurrence or non-occurrence of another event.

Similarly, the following connotation also holds true.

P(B|A) = P(B)

It means that if A and B are two independent events, the probability of event B, given that event A occurs, is equal to the probability of event B.

Further, there is one more observation that is true for such events.

P(A and B) = P(A) * P(B)

The above equation suggests that if events A and B are independent, the probability of both the events occurring is equivalent to the product of their individual probabilities.

Independent Events in Probability

In the terminology of probability, two events can be said to independent if the outcome of one event is not decisive of the probability of occurrence or non-occurrence of another event.

Following is the calculation of probability for any event –

For eg:
Source: Independent Events (wallstreetmojo.com)

For example, let us calculate the probability of getting 6 on the dice when we roll it. Here, the total number of outcomes is six (numbers 1,2,3,4,5 and 6), and a number of favorable outcomes are one (number 6). Hence, probability comes out to be 0.16.

Independent vs. Dependent Events

• Two events are said to be independent when the probability of one event does not impact the probability of another event. For example, simultaneously tossing two coins are independent events because the probability of head or tail on the first coin is not dependent or decisive of the probability of head or tail on another coin.
• On the other hand, two events are called dependent if the outcome of one of the events can alter the probability of another event. In simple terms, when the outcome of one event can influence the occurrence of another event, the events are said to be dependent events. For example, in a deck of 52 cards, two cards are chosen randomly one by one. Now, if the first card is chosen and it is not replaced, the probability of the second card will definitely change since after the first card is removed, only 51 cards are to remain in the deck. It results in the two events being dependent events.

Conclusion

For concluding whether the events are dependent or not, one needs to analyze whether the occurrence of one event may alter the probability of occurrence of the second event. One may calculate the probability of both the events and apply multiplication rules to test the independence test.

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