Understanding Probability: Joint Tables, Conditional Probability, and Independence

Exam Tip: Always check your "Total" row and "Total" column. The sum of all individual joint probabilities in the table must equal 1.0.

Calculating Probabilities of Single and Combined Events

Once your table is constructed, you can answer complex questions about specific groups or combined occurrences.

Probability of a Single Event

To find the probability of a single event (the "marginal probability"), you simply look at the total for that category. For instance, in the e-commerce example where 600 visitors were tracked, we can determine the probability of a visitor viewing a redesign ($R$) or completing a purchase ($P$).

The "Or" Rule: Probability of Redesign or Purchase

When asked for the probability that a visitor saw the redesign or made a purchase, we use the Addition Rule. We cannot simply add the two individual probabilities because that would count the people who did both twice.

The formula is:

$$\begin{aligned} P(A \cup B) = P(A) + P(B) - P(A \cap B) \end{aligned}$$

Using the e-commerce data:

• $P(\text{Redesign}) = 0.5$
• $P(\text{Purchase}) = 0.25$
• $P(\text{Redesign and Purchase}) = 0.15$

The calculation follows:

$$\begin{aligned} P(\text{Redesign or Purchase}) = 0.5 + 0.25 - 0.15 = 0.6 \end{aligned}$$

Mastering Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already happened. This effectively "shrinks" your universe of interest from the entire dataset to a specific row or column.

The notation $P(A \mid B)$ is read as "the probability of A given B." The formula is:

$$\begin{aligned} P(A \mid B) = \frac{P(A \cap B)}{P(B)} \end{aligned}$$

Worked Examples

1. Marketing Survey: If a consumer ranked a product as "fair," what is the probability they had "high awareness"? We look only at the "Fair" row in the survey table and divide the "Fair + High Awareness" cell by the total "Fair" row.
2. E-commerce: What is the probability that a visitor who saw the redesign completed a purchase? $$\begin{aligned} P(\text{Purch} \mid \text{Red}) = \frac{0.15}{0.5} = 0.3 \end{aligned}$$ Conversely, for those who saw the old design: $$\begin{aligned} P(\text{Purch} \mid \sim\text{Red}) = \frac{0.1}{0.5} = 0.2 \end{aligned}$$

Determining if Events are Independent

The most critical question in data analysis is often: "Does one event affect the other?" In probability, two events are independent if the occurrence of one does not change the probability of the other.

The Test for Independence

To mathematically prove independence, you check if the probability of both events happening together is equal to the product of their individual probabilities:

$$\begin{aligned} P(A \cap B) = P(A) \cdot P(B) \end{aligned}$$

If this equation holds true, the events are independent. If not, they are dependent (or "associated").

Applying the Test

Let's return to the e-commerce redesign study:

• $P(\text{Purchase}) = 0.25$
• $P(\text{Redesign}) = 0.5$
• $P(\text{Purchase and Redesign}) = 0.15$

First, calculate the product of the individual probabilities:

$$\begin{aligned} P(\text{Purch}) \cdot P(\text{Red}) = 0.25 \cdot 0.5 = 0.125 \end{aligned}$$

Now, compare this to the actual joint probability:

$$\begin{aligned} 0.15 \neq 0.125 \end{aligned}$$

Because $P(A \cap B) \neq P(A) \cdot P(B)$, the events are not independent. In a business context, this suggests that seeing the redesign actually influenced the likelihood of a purchase.

Summary Checklist:

• Mutually Exclusive: Events cannot happen at the same time ($P(A \cap B) = 0$).
• Collectively Exhaustive: The events cover all possible outcomes ($P(A \cup B) = 1$).
• Independent: One event doesn't change the odds of the other ($P(A \cap B) = P(A) \cdot P(B)$).