Essentials of Data Science – Probability and Statistical Inference – Independent Events

By / February 10, 2022

In the previous note on probability and statistical inference, we have seen axioms of probability, counting principles, conditional probability, multiplication theorem of probability, total probability and Bayes theorem. In this note, we will learn and understand the Independent events and how it is different from dependent events.

Let us understand independent events with a simple analogy of human life and relationship.

Two persons are said to be independent if not dependent on each other, in the sense that the occurrence or non-occurrence of one does not affect another. On the other way round, an example of a couple can’t be considered independent as both directly or indirectly affect each other. It means there is some association or relationship between the two individuals.

Similarly, in probability and statistics, when there is no association or relationship between two events, those events are called independent events.

Table of Contents hide
1 Independent Events
1.1 Generalized for n events
2 References
2.1 Share this:
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Independent Events

Two events are independent if the occurrence or non-occurrence of one event does not affect the occurrence or non-occurrence of the other event. In other words, two events A and B, are independent if the probability of occurrence of B does not affect the likelihood of occurrence of A.

For example, in conditional probability, we have seen that events are dependent, which means the occurrence of B influences the occurrence of A. However, if the occurrence of B does not influence the occurrence of A, then we can say that the two events are independent, and it is illustrated as follows:

P(A|B) = P(A) and P(A| \bar{B}) = P(A) which implies P(A \cap B) = P(A) \times P(B)

The random events A and B are called independent if P(A \cap B) = P(A) \times P(B). It means, the probability of simultaneous occurrence of both events A and B is the product of A and B’s probabilities of occurrence.

Generalized for n events

The n events A_1, A_2, A_3, \cdots , A_n are stochastically mutually independent, if for any subset of m events A_{i1}, A_{i2}, \cdots, A_{im} where (m \leq n) such that:

P(A_{i1} \cap A_{i2} \cdots A_{im}) = A_{i1} \times A_{i2} \cdots \times A_{im}

A weaker form of independence is called pairwise independence. The above condition is fulfilled only for two random events, i.e., m = 2, so mutually independence implies pairwise independence; however, the converse may not hold.

Example 1: An ordinary deck has 52 playing cards and suppose one card is selected at random from it. If K is an event of selected card is a king and H is an event that the selected card is a heart. In this example, we will show that these two events are not independent.

Solution:

  • P(K) = P( Selected card is a king) = 4/52, as there are total of 4 king in an ordinary deck of 52 cards. So the selection of one king card is 4/52.
  • P(H) = P(Selected card is a heart) = 13/52, as there are total of 13 heart cards and one of the card is king as well.

If two events are independent then P(K \cap H) = P(K) \times P(H) = 4/52 x 13/52 = 1, where as the actual answer is 1/52.

References

  1. Essentials of Data Science With R Software – 1: Probability and Statistical Inference, By Prof. Shalabh, Dept. of Mathematics and Statistics, IIT Kanpur.

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