Dependent Event vs Independent Event

Dependent events and independent events are opposing events. In probability theory, a set of independent events occur independently of each other and are not influenced by the outcome of other events. On the other hand, events in a set of dependent events are influenced by the outcome of preceding events. In other words, dependent events are linked, where the chance of one event happening affects the probability of the subsequent event occurring or not occurring.

• Dependent events vs. independent events explain that independent events have separate and unrelated outcomes, while dependent events are connected.
• Independent events are those in which one event's outcome does not influence another event's outcome. Whereas, dependent events are those in which another event influences the occurrence or non-occurrence of one event; they are interconnected, and one event affects the probability of the other event happening.
• Events are categorized as dependent or independent in real-world scenarios and mathematical analyses based on their relationship.
• Various fields such as business, finance, healthcare, and everyday life experience, both dependent and independent events, showcase their significance across different domains.

Here are the main differences between the two:

A dependent event is an occurrence that depends on or has happened because something else or another event happened, which led to this event. Dependent events are never separate events but are mostly consequences of what has happened previously, which eventually made the event happen. Such events are outcomes and impact a statistical analysis or day-to-day life differently.

For a mathematical dependent event example, suppose a pouch contains two pink, four black, and three white balls. When a person randomly pulls out one ball from the pouch and is not replaced with any other ball and again removes a ball from the pouch, what are the chances that the first ball was pink and the second ball was black?

Now, since the first ball was removed and then not replaced, the sample size of the pouch is different for both the first and second ball; for the first ball, it is 9, and for the second ball, it is 8, which itself denotes that both events are dependent.

• P(A), the probability of drawing a pink ball on the first draw
• P(A)= 2/9
• P(B∣A), the probability of drawing a black ball on the second draw, given that a pink ball was drawn on the first draw and not replaced
• P(B∣A)= 4/8

• The overall probability of drawing a pink ball on the first draw and a black ball on the second draw:
• P(A and B)=P(A)×P(B∣A)

=2/9 x 4/8

=1/9

In the real world, there can be many day-to-day examples of dependent events, such as studying for an exam and passing the test, parking in the no parking area and receiving a ticket for it, etc. These are simple dependent event examples where the second event is the outcome of the first event.

Independent events are separate events with absolutely no interference, influence, or impact or any other event occurring or not occurring. These can be two or more events that occur simultaneously or consecutively, irrespective of whether a different consequence occurs. These events are first-hand experiences or happenings and are not the result of any other event. Two independent events can co-exist without influencing each other and processing individually.

Again, for a mathematical independent event example, suppose the same pouch with the same contents, two pink balls, four black balls, and three white balls; this time, when a person randomly pulls out one ball from the pouch, replace it and again removes a ball from the pouch, what are the chances that the first ball was pink and the second ball was black?

• P(A), the probability of drawing a pink ball on the first draw
• P(A)= 2/9
• P(B), the probability of drawing a black ball on the second draw, given that a pink ball was drawn on the first draw and replaced
• P(B)= 4/9

• The overall probability of drawing a pink ball on the first draw and a black ball on the second draw:
• P(A and B)=P(A)×P(B)

=2/9 x 4/9

=8/81

It is well observed that in both the examples, the situation was the same, the balls were the same, and the final objective was the same, but due to a slight change, the outcome of both the examples is different.

Apart from a mathematical example, many independent events happen in daily life, such as reading a book and going on a vacation, cooking a good meal, and getting a promotion at work. All these are independent events with absolutely no impact on one another.

Here are the similarities between the two:

• Both dependent and independent events share the same equation, determining the condition P(B|A) = P(B). P(B∣A) represents the conditional probability of event B, given event A has occurred, and P(B) represents the probability of event B occurring independently of event A.
• If the condition holds true, A and B are designated as independent events.
• Both events can be tested mathematically.
• Both dependent and independent events can be observed in normal day-to-day life.

This article has been a guide to Dependent Events vs Independent Events. Here, we explain their differences, a comparative table, and similarities. You may also find some useful articles here -

• Independent Variable
• Independent Project
• Dependent Variable