Determine whether events are independent based on conditional probabilities (SAT). | Probability and conditional probability | SAT Math

Check independence by comparing $𝑃 (𝐴 ∣ 𝐵)$ with $𝑃 (𝐴)$ , or by checking $𝑃 (𝐴 a n d 𝐵) = 𝑃 (𝐴) 𝑃 (𝐵)$ .

Core Idea

Two events are independent when knowing one occurred doesn't change the probability of the other. Check whether $𝑃 (𝐴 ∣ 𝐵) = 𝑃 (𝐴)$ ; if not, the events are dependent.

Understanding

Independence is a specific relationship: event A is independent of event B when

$𝑃 (𝐴 ∣ 𝐵) = 𝑃 (𝐴)$

In words: learning that B happened gives you no new information about A.

If $𝑃 (𝐴 ∣ 𝐵) \neq 𝑃 (𝐴)$ , the events are dependent — knowing B changes your estimate of A.

On the SAT, you'll usually be given a two-way table and asked whether two categories are independent. The test:

Compute $𝑃 (𝐴)$ using the grand total.
Compute $𝑃 (𝐴 ∣ 𝐵)$ using the B subtotal.
Compare. Same? Independent. Different? Dependent.

Equivalently, you can check $𝑃 (𝐴 a n d 𝐵) = 𝑃 (𝐴) \cdot 𝑃 (𝐵)$ . This is the same test, just rearranged.

A common trap: students assume that two events being "unrelated in real life" means they're independent. On the SAT, independence is a mathematical property you verify with numbers, not intuition.

Step by Step

Compute $𝑃 (𝐴)$ from the overall totals.
Compute $𝑃 (𝐴 ∣ 𝐵)$ from the B subgroup.
If $𝑃 (𝐴 ∣ 𝐵) = 𝑃 (𝐴)$ , the events are independent. If they differ, the events are dependent.
Alternatively, check whether $𝑃 (𝐴 a n d 𝐵) = 𝑃 (𝐴) \times 𝑃 (𝐵)$ .

Misconceptions

Assuming events are independent because they seem unrelated in everyday reasoning.
Confusing independent with mutually exclusive — mutually exclusive events (can't both happen) are actually dependent unless one has probability 0.
Rounding both probabilities and then claiming they're equal when exact fractions differ.

Question

Worked Example

A school surveyed 300 students about pet ownership and instrument playing. 120 students own a pet, 90 play an instrument, and 36 both own a pet and play an instrument. Based on this data, are the events "owns a pet" and "plays an instrument" independent?

Select an answer to see the explanation