Dependent and Independent Events

Dylan | Sep 16, 2019

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In the field of Probability, events can only have two fundamental types of relationships. The events are either independent or dependent. When we begin to consider conditional probability, It is very important to be able to determine and identify the relationship between events.

This is due to the fact that each relationship results in different expected behaviors. If a relationship is independent, knowing that one event occurred does not influence our beliefs about the second event in the way that it would in a dependent relationship. This distinction is made clearer by the different formulas and equations used to handle the two types of relationships.

Let’s begin by exploring event Independence.

Independence

When two events are independent of one another, whether or not one event occurred it does it change any of our beliefs about the other event.

On the surface, this may seem confusing but the following example should help clear it up.

Imagine you have a two-sided fair coin. In this case, fair refers to the fact that when flipped, each side of the coin is equally likely to land face-up. Obviously, the probability of the coin landing Head’s side up is 50%. You flip the coin once and it happens to land Head’s up. Now, knowing the result of this coin flip, what is the probability that the second coin toss will be Head’s up as well?

It’s easy to be tricked by your intuition to thinking that the next flip is more likely to be Tail’s side up because the first was Head’s side up but tread carefully! Does knowing the result of the previous coin toss really impact the probability of the following coin toss? Could the previous Head’s up toss possibly change the “fairness” of our coin? The answer is no.

Our coin remains fair. For any individual toss, the probability that it will be Heads is 50% and the probability it will be Tails is an equal 50%. Knowing the outcome of the previous toss doesn’t change this fact. This is a clear example of event independence!

So, the results of multiple coin tosses are independent of each other but how can we prove it? Mathematically, the events A and B are independent if P(A∩B) = P(A) x P(B). In our case, let’s pretend that event A is getting Heads and event B is getting Tails. We know that both P(A) = 0.5 and P(B) = 0.5 and when we multiply them, 0.5 x 0.5 = 0.25.

Let’s see if 0.25 is equal to P(A∩B). Let’s take a look at the chart below.

Probability Tree

According to the chart, the chances of flipping Heads is 0.5 (exactly as we would expect). So, the probability of getting a Heads followed by a Tails is 0.25 because after flipping a coin two times, there are only 4 possible sequences: (Heads, Tails), (Heads, Heads,), (Tails, Tails), and (Tails, Heads). Since we know the coin is fair, each outcome has an equal probability of 0.25 of occurring.

This confirms our assertion of event independence because 0.25 is indeed equal to 0.25! Now, what about dependence?

Dependence

Dependence is the opposite of independence, by knowing that one even occurred in a dependent relationship, our beliefs about whether the second event occurred or not are influenced.

Returning to the coin, a simple example of dependence is the event that a single coin toss results in Heads and the equally likely event that the same coin toss resulted in Tails. When we toss a coin, there are only two possible outcomes. In this case, because the relationships are dependent, knowing that one event definitely occurs changes our beliefs about the occurrence (or lack of) of the second event.

In other words, when we flip the coin and see that the coin landed Head’s up because the two outcomes of the coin toss are dependent on one another, we can conclude that the event where the coin landed Tails did not occur. By knowing how the coin landed, we gain information about the other possible events.

Any event that is not independent and therefore violates the equation P(A∩B) = P(A) x P(B) is automatically by default dependent.

Thanks for reading and I sincerely hope you were able to learn something about event dependence and independence in probability! As always, if anything remains unclear, please ask your questions in the comments below and I look forward to exploring the topic further! Thanks again and see you again next week!

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