Probability • Topic 4 of 4

Complementary Events

The complement of an event E is "E does not happen", and their probabilities always sum to 1, so P(not E) = 1 − P(E). This tiny rule is one of the most powerful shortcuts on the GRE, because many questions are far easier to answer through the back door. The flagship case is "at least one": rather than adding the messy probabilities of exactly one, exactly two, and so on, compute P(at least one) = 1 − P(none). The probability of at least one head in 4 coin tosses is 1 − (1/2)⁴ = 1 − 1/16 = 15/16 — one line instead of four. The same move rescues "at least one defective", "at least one match", and "at least one success" problems across the test. Whenever you see "at least", train yourself to reach for the complement first; the direct count is almost always the slower, error-prone path.

✅ Solved examples

1. The probability of rain tomorrow is 0.3. Probability it does not rain?
Complement: 1 − 0.3 = 0.7.
2. A coin is tossed 3 times. Probability of at least one head?
Complement of "no heads" (all tails): P(TTT) = (1/2)³ = 1/8. So P(at least one head) = 1 − 1/8 = 7/8.
3. A die is rolled twice. Probability of at least one 6?
P(no 6 on a roll) = 5/6, so P(no 6 twice) = (5/6)² = 25/36. P(at least one 6) = 1 − 25/36 = 11/36.
4. Quantitative Comparison. A fair coin is tossed 4 times. Quantity A: P(at least one tail). Quantity B: 14/15. Which is greater?
P(no tail) = P(all heads) = (1/2)⁴ = 1/16. P(at least one tail) = 1 − 1/16 = 15/16 = 0.9375. Quantity B = 14/15 ≈ 0.9333. Quantity A is greater.

✏️ Practice — try these, take hints as needed

1. If P(winning) = 2/7, what is P(not winning)?
Complements sum to 1.
1 − 2/7.
Subtract.
5/7
2. A die is rolled. Probability of NOT rolling a 3?
P(3) = 1/6.
1 − 1/6.
Subtract.
5/6
3. A coin is tossed twice. Probability of at least one head?
P(no head) = P(TT) = 1/4.
1 − 1/4.
Subtract.
3/4
4. A die is rolled 3 times. Probability of at least one 6?
P(no 6 each roll) = 5/6.
P(no 6 all three) = (5/6)³ = 125/216.
1 − 125/216.
91/216
5. A bag has 10 items, 2 defective. One is drawn. Probability it is NOT defective?
P(defective) = 2/10.
1 − 1/5.
Subtract.
4/5

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Mutually Exclusive & "Or" Take the chapter test →