Independent vs. Dependent Events

In the previous section, you learned about conditional probability, $P(A|B)$, which tells us the probability of event $A$ happening given that event $B$ has already occurred. Now, we'll look at a special relationship between events: independence and dependence. Understanding this distinction is significant when analyzing data and building certain types of machine learning models.

What Does Independence Mean?

Two events, $A$ and $B$, are considered independent if the occurrence of one event does not affect the probability of the other event occurring. Think about flipping a fair coin twice. Does the outcome of the first flip (say, getting Heads) change the probability of getting Heads on the second flip? No, the probability remains $1/2$. The coin has no memory.

Formally, we can define independence in two main ways:

1. Using Conditional Probability: Events $A$ and $B$ are independent if knowing $B$ occurred doesn't change the probability of $A$.

   $$P(A|B) = P(A)$$

   Similarly, if knowing $A$ occurred doesn't change the probability of $B$:

   $$P(B|A) = P(B)$$

   (Assuming $P(B) > 0$ for the first case and $P(A) > 0$ for the second).

2. Using Joint Probability: Events $A$ and $B$ are independent if the probability of both events occurring together is simply the product of their individual probabilities.

   $$P(A \cap B) = P(A) P(B)$$

   This is often the most practical way to check for independence. If this equation holds true, the events are independent. Otherwise, they are dependent.

Example: Coin Flips

Let's revisit the two coin flips. Event A: Getting Heads on the first flip. $P(A) = 1/2$. Event B: Getting Heads on the second flip. $P(B) = 1/2$.

What's the probability of getting Heads on both flips, $P(A \cap B)$? The possible outcomes are HH, HT, TH, TT. Only HH satisfies both events. So, $P(A \cap B) = 1/4$.

Now let's check using the formula: $P(A) P(B) = (1/2) \times (1/2) = 1/4$.

Since $P(A \cap B) = P(A) P(B)$, the two events are independent, just as our intuition suggested.

What Does Dependence Mean?

If the occurrence of one event does affect the probability of another event occurring, the events are dependent.

Consider drawing two cards from a standard 52-card deck without replacing the first card. Event A: Drawing a King on the first draw. Event B: Drawing a King on the second draw.

Are these events independent? Let's think about the probabilities. $P(A) = 4/52 = 1/13$ (There are 4 Kings in 52 cards).

Now, what is the probability of drawing a King on the second draw, given that we drew a King on the first draw? This is $P(B|A)$. If we drew a King first, there are now only 3 Kings left and only 51 total cards remaining. $P(B|A) = 3/51 = 1/17$.

Since $P(B|A) = 1/17$ and $P(A) = 1/13$ (and intuitively, $P(B)$ would also be $1/13$ if we didn't know the first card's outcome), we see that $P(B|A) \neq P(B)$. Knowing the outcome of the first draw changed the probability for the second draw. Therefore, the events are dependent.

We can also see this using the joint probability definition. We know from the conditional probability formula ($P(A \cap B) = P(B|A)P(A)$): $P(A \cap B) = P(\text{King first AND King second}) = P(B|A)P(A) = (3/51) \times (4/52) = 12/2652 \approx 0.0045$

Now compare this to the product of their individual probabilities: $P(A)P(B) = (4/52) \times (4/52) = (1/13) \times (1/13) = 1/169 \approx 0.0059$

Since $P(A \cap B) \neq P(A)P(B)$, the events are confirmed to be dependent. The act of drawing without replacement links the outcomes. If we had replaced the first card, the events would have been independent.

Why Independence Matters in Machine Learning

The concepts of independence and dependence are fundamental in many areas of machine learning:

1. Feature Engineering: When selecting features for a model, understanding if features are dependent can be important. Highly dependent features might be redundant, providing similar information. Sometimes, combining dependent features or removing one can improve model performance or reduce complexity.
2. Model Assumptions: Some models make explicit assumptions about independence. A classic example is the Naive Bayes classifier. It works by assuming that all input features are independent of each other, given the class label. This is a "naive" assumption because features in real-world data are often dependent. However, this simplification makes the calculations much easier and the model surprisingly effective in many cases (like text classification). Knowing about independence helps you understand the underlying assumptions and potential limitations of such models.
3. Probability Models: When building probabilistic models (like Bayesian networks), the dependencies between variables are explicitly mapped out. Independence allows for simplifications in the model structure and calculations.

In summary, distinguishing between independent and dependent events allows us to correctly calculate probabilities involving multiple events and to understand the assumptions behind certain machine learning algorithms. Independence simplifies calculations ($P(A \cap B) = P(A)P(B)$), while dependence requires using conditional probabilities ($P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$).