Independence of Events

In probability theory and statistics, two events are independent if the occurrence of one event does not affect the chances of the occurrence of the other event. The mathematical formulation of the independence of events A and B is the probability of the occurrence of both A and B being equal to the product of the probabilities of A and B (i.e., P(A and B) = P(A)P(B)).

This blog post will explore the concept of independence of events in more detail and provide some examples to illustrate how it works in practice.

Independent vs. Dependent Events

Most people intuitively understand what it means for two events to be independent. For example, consider the following two events:

• Event A: It rains tomorrow.
• Event B: I take my umbrella with me tomorrow.

It is clear that these two events are not independent, because if it rains tomorrow, then the chances of me taking my umbrella with me tomorrow go up. Conversely, if I know that I am going to take my umbrella with me tomorrow, then that means it is more likely that it will rain tomorrow. Formally, we would say that these events are dependent.

On the other hand, consider these two events:

• Event C: I roll a die and get a 6.
• Event D: I roll a die again and get a 4.

These events are independent, because the occurrence of one event does not affect the chances of the other event occurring. That is, whether or not I rolled a 6 on my first die roll does not affect my chances of rolling a 4 on my second die roll.

Independence vs. Correlation

It is important to note that independence and correlation are not the same thing. Two variables can be correlated without being dependent. For example, imagine that you are trying to predict daily stock market returns using data on daily temperatures. You would likely find that there is a positive correlation between these two variables—that is, as temperatures go up, stock market returns tend to go up as well—but clearly they are not dependent on each other (a heat wave is not going to cause the stock market to crash!). All this is to say that correlation does not imply causation: just because two variables are correlated does not mean that one variable causes the other variable to change.

Independent vs Mutually Exclusive Events

Another concept that is often confused with independence is mutually exclusive events. Mutually exclusive events are two events that cannot happen at the same time—for example, flipping a coin and getting a head OR flipping a coin and getting a tail but never getting both at once. Independent events are two events where the occurrence of one event does NOT affect whether or not the other event occurs; mutually exclusive events are those where one event MUST preclude the other from happening (they cannot occur simultaneously). So while independent events are not affected by each other, mutually exclusive events cannot even co-occur!

In conclusion, independence of events means that the occurrence of one event does NOT affect whether or not another event occurs; mutual exclusive events cannot even co-occur! Keep this distinction in mind as you further your studies in probability theory and statistics!