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Digital SAT · Probability & Two-Way Tables

How to Solve Complement and Total Probability on the Digital SAT

Something has to happen, so all the probabilities in a situation add to one. The complement rule leans on that: the chance an event does not occur is one minus the chance it does. When several outcomes are listed with one blank, subtract the known ones from one to fill it. As long as the outcomes cover everything without overlap, the total stays at one.

Written from Perfect1600’s analysis of every complement and total probability question in our bank·Method checked against the current Bluebook test
per test
less than 1 per test
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Medium
typical difficulty
practice questions
34
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

1 − P(A)
Everything sums to one

The complement is one minus the event; a missing outcome is one minus the rest.

subtract
No calculator

A subtraction from one, not a graphing task.

reread
Event vs complement

Confusing the event with its complement is the frequent slip.

How to recognize complement and total probability questions

  • Several outcomes are given with probabilities, and one is missing.
  • The question asks for the chance an event does not happen.
  • The words at least, not, or complement appear.
  • Choices are fractions, decimals, or percents.

Why students miss these

The idea is easy to misapply. People subtract from the wrong total, forget the listed outcomes must cover everything, or compute the event instead of its complement. The at-least phrasing often points to the complement, since subtracting the one case you do not want is quicker. Checking that all the probabilities sum to one keeps it honest.

The step-by-step method

  1. 1

    Confirm full coverage

    Make sure the listed outcomes are all the possibilities and do not overlap.

  2. 2

    Add the known probabilities

    Sum the probabilities you are given.

  3. 3

    Subtract from one

    The missing probability, or the complement, is one minus that sum.

  4. 4

    Match the asked event

    Confirm you reported the event or its complement, whichever was wanted.

Worked examples

Medium example

A jar contains 80 beads: 20 are red, 12 are blue, and the rest are yellow.

If one bead is selected at random, what is the probability of selecting a bead that is red or blue?
A
0.15
B
0.25
0.40
D
0.60

A: Incorrect. This is the probability of selecting a blue bead.

B: Incorrect. This is the probability of selecting a red bead.

C: Correct. Red or blue totals 20+12=3220 + 12 = 32, so the probability is 3280=0.40\frac{32}{80} = 0.40.

D: Incorrect. This is the probability of selecting a yellow bead, the complement of red or blue.

Explanation

Red or blue: 20+12=3220 + 12 = 32; probability 3280=0.40\frac{32}{80} = 0.40.

Detailed explanation

Red or blue totals 20+12=3220 + 12 = 32 out of 80, so the probability is 3280=0.40\frac{32}{80} = 0.40.

Medium example

A bag contains 50 balls: 12 are red, 18 are blue, and the rest are green.

If one ball is selected at random, what is the probability of selecting a green ball?
A
0.24
B
0.36
0.40
D
0.60

A: Incorrect. This is the probability of selecting a red ball.

B: Incorrect. This is the probability of selecting a blue ball.

C: Correct. Green is 50−12−18=2050 - 12 - 18 = 20, so the probability is 2050=0.40\frac{20}{50} = 0.40.

D: Incorrect. This uses the combined red and blue count instead of the green count.

Explanation

Green: 50−30=2050 - 30 = 20; probability 2050=0.40\frac{20}{50} = 0.40.

Detailed explanation

Subtracting the known parts gives 50−12−18=2050 - 12 - 18 = 20 green, so the probability is 2050=0.40\frac{20}{50} = 0.40.

Medium example

A spinner has four outcomes. The probabilities of three of them are P(A)=0.15P(A) = 0.15, P(B)=0.25P(B) = 0.25, and P(C)=0.30P(C) = 0.30.

What is the probability of the fourth outcome, D?
A
0.20
0.30
C
0.40
D
0.70

A: Incorrect. This subtracts the known parts from 0.90 instead of 1.

B: Correct. P(D)=1−(0.15+0.25+0.30)=0.30P(D) = 1 - (0.15 + 0.25 + 0.30) = 0.30.

C: Incorrect. This subtracts only two of the three known probabilities.

D: Incorrect. This is the sum of the three known probabilities, not the remaining one.

Explanation

Remaining: 1−0.70=0.301 - 0.70 = 0.30.

Detailed explanation

The four probabilities sum to 1, so P(D)=1−(0.15+0.25+0.30)=0.30P(D) = 1 - (0.15 + 0.25 + 0.30) = 0.30.

The common traps

PatternWhat it doesThe tell
Computed the event, not the complementGave the event's probability when its complement was asked.The complement is one minus the event; reread which is wanted.
Incomplete coverageAssumed a complete list when an outcome was missing.The probabilities must sum to one across all outcomes.
Wrong totalSubtracted from something other than one.Total probability is always one.

Try it: two real questions

Question 1hard

A spinner has three outcomes, A, B, and C. The probability of outcome A is 0.2, and the probability of outcome B is twice the probability of outcome A.

What is the probability of outcome C?

Question 2hard

A bag contains red, blue, and green tokens. The probability of drawing a red token is 13\frac{1}{3}, and the probability of drawing a blue token is 14\frac{1}{4}.

What is the probability of drawing a green token?

Question 3 is ready when you are

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Often confused with

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Probability & Two-Way Tables
Probability from a Two-Way Table
Percentages
Percentages

Related reading

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Which SAT skills move your score most

Common questions

What is the complement rule in probability?

The chance an event does not occur is one minus the chance it does. Since all outcomes sum to one, subtracting the event's probability from one gives its complement.

How do I find a missing probability in a list?

Add the probabilities you are given and subtract from one. Because the outcomes cover the whole situation, the missing one makes the total equal one.

When should I use the complement?

Often with at-least or not phrasing. It can be faster to find the one case you do not want and subtract it from one than to add many cases.

Why do all the probabilities add to one?

Because some outcome must occur. When the outcomes cover every possibility without overlap, their probabilities together account for the whole situation, which is one.

Practice complement and total probability the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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