L2 · Complements: The Probability of NOT

Learning goal

“I can find the probability of a complement - P(A^c) = 1 - P(A) - recognize complementary events, and move between counting the 'not' region directly and computing it from the event.”

Complement (A^c) — the event consisting of all outcomes in the sample space that are NOT in A. Read "A-complement" or "not A."

Complementary events — a pair of events that are complements of each other: they cannot both occur, and one of them must occur (e.g. rolling a 4, and rolling a number other than 4).

Sample space — the set of all possible outcomes; the complement is always measured against it. A and A^c together rebuild it exactly.

Worked example

A bag holds 4 red, 3 blue and 5 green marbles (12 in total). One marble is drawn. Find P(not red). Method 1 (the rule): P(red) = 4/12, so P(not red) = 1 - 4/12 = 8/12 = 2/3. Method 2 (counting): "not red" = blue or green = 3 + 5 = 8 marbles, so P(not red) = 8/12 = 2/3. The two methods agree because every marble is either red or not red - the two events split the sample space with no overlap and nothing left over. That is why P(A) + P(A^c) = 1 for every event A. Say it plainly: when "not" is easier to count than the event itself - or the other way around - the complement rule lets you work whichever side is cheaper.

Check yourself — explain it

Why does P(A) + P(A^c) always equal exactly 1?

Paired problem — example still visible

A spinner has 10 equal sections numbered 1 to 10. Find P(not landing on 7), by the rule and by counting. -> Rule: 1 - 1/10 = 9/10. Counting: 9 of the 10 sections are not 7 -> 9/10

Faded completion — last step is yours

"The probability of rain tomorrow is 0.28. P(no rain) = 1 - ___ = ___." -> 1 - 0.28 = 0.72

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