Apply complement and total probability ideas (sum of probabilities is 1). | Probability and conditional probability | SAT Math

Use the complement when counting the event directly is harder.

Core Idea

The probability of an event and the probability of its complement always sum to 1. When counting the event directly is hard, subtract from 1 instead.

Understanding

Every event splits the sample space into two parts: the event happening and the event not happening. Together they cover everything, so

$𝑃 (𝐴) + 𝑃 (n o t 𝐴) = 1$

This gives you the complement rule:

$𝑃 (n o t 𝐴) = 1 - 𝑃 (𝐴)$

This is most useful when the "not" side is simpler to count. For instance, if you need $𝑃 (a t l e a s t o n e h e a d s i n 4 f l i p s)$ , it's much easier to compute $𝑃 (n o h e a d s) = (12) 4 = 116$ , then subtract: $1 - 116 = 1516$ .

The SAT also uses the fact that all probabilities in a complete set of outcomes sum to 1. If a spinner has four regions with probabilities 0.3, 0.25, 0.15, and $𝑥$ , then $𝑥 = 1 - 0.3 - 0.25 - 0.15 = 0.3$ .

Watch for problems that phrase complements indirectly: "not red" means all other colors combined; "fails to win" means loses or draws.

Step by Step

Determine whether counting the event directly or counting its complement is easier.
If the complement is easier, compute $𝑃 (c o m p l e m e n t)$ first.
Subtract from 1: $𝑃 (e v e n t) = 1 - 𝑃 (c o m p l e m e n t)$ .
When given partial probabilities that must total 1, set up the equation and solve for the unknown.

Misconceptions

Forgetting that 'at least one' is best handled via the complement 'none' — students try to count every case individually.
Treating 'not A' as a single specific outcome rather than everything outside A.
Assuming probabilities must sum to 1 within a row or column of a two-way table — they sum to 1 only when the row or column represents a complete partition.

Question

Worked Example

A bag contains red, blue, and green marbles. The probability of randomly selecting a red marble is 0.35 and the probability of selecting a blue marble is 0.25. What is the probability of selecting a green marble?

Select an answer to see the explanation