知识点 4

Determine whether events are independent based on conditional probabilities (SAT).

Check independence by comparing 𝑃(𝐴 ∣𝐵) with 𝑃(𝐴), or by checking 𝑃(𝐴 and 𝐵) =𝑃(𝐴)𝑃(𝐵).

核心知识

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.

深入理解

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) ≠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:

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

Equivalently, you can check 𝑃(𝐴 and 𝐵) =𝑃(𝐴) ⋅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.

分步讲解

  1. Compute 𝑃(𝐴) from the overall totals.
  2. Compute 𝑃(𝐴 ∣𝐵) from the B subgroup.
  3. If 𝑃(𝐴 ∣𝐵) =𝑃(𝐴), the events are independent. If they differ, the events are dependent.
  4. Alternatively, check whether 𝑃(𝐴 and 𝐵) =𝑃(𝐴) ×𝑃(𝐵).

常见误解

  • 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.
题目

示例解析

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?

选择一个答案查看解析