Math Vale

🎲 Probability Marsh · Probability

Complements, Or & And

Combine probabilities: the complement rule for "not", the addition rule for "or", and the multiplication rule for independent events.

In short

  • P(not A) = 1 - P(A), and it turns most "at least one" questions into a single multiplication.
  • "Or" adds, but the overlap is subtracted once when the two events can both happen: P(A or B) = P(A) + P(B) - P(A and B).
  • "And" multiplies for independent events, which is why the answer is always smaller than either probability alone.
  • Before computing, decide which of the three rules the wording asks for — the arithmetic is the easy part.

Not: the complement rule

Every event has an opposite, called its complement. Either the event happens or it does not, and there is no third possibility — so between them they cover everything, and everything has probability 1.

P(not A) = 1 - P(A)

If the probability of rain is 3/10, the probability of no rain is 7/10. If P(a six) = 1/6, then P(not a six) = 5/6.

This tiny rule does far more work than it looks like it should, because a great many awkward questions have an easy opposite. "At least one" is the classic: listing all the ways to get at least one six out of three rolls is slow, but there is exactly one way to fail — no sixes at all — so work that out and subtract from 1.

Or: the addition rule

"Or" in mathematics is generous: A or B means A, or B, or both.

When two events cannot both happen — drawing a token that is red or blue, rolling a 2 or a 5 — the favourable outcomes simply pile up, so the probabilities add:

P(A or B) = P(A) + P(B)

When they *can* both happen, adding double-counts the overlap. If 6 ferry passengers carry a lantern and 6 carry a map, but 2 carry both, then 6 + 6 = 12 counts those 2 people twice. There are really 10 people carrying at least one:

P(A or B) = P(A) + P(B) - P(A and B)

The first version is just the second version with an overlap of zero, so if you only remember one rule, remember the one with the subtraction in it.

And: the multiplication rule

"And" asks for both things to happen, which is more demanding than asking for either — so the answer must come out smaller than each piece. That is what multiplying does to numbers under 1.

Two events are independent when the first tells you nothing about the second: two coin tosses, a die and a spinner, a draw with the token put back. For independent events:

P(A and B) = P(A) x P(B)

P(heads) x P(a six) = 1/2 x 1/6 = 1/12, which matches the 12-cell grid exactly — one favourable cell out of twelve.

Adding here is the classic error, and it fails a sanity check instantly: 1/2 + 1/6 = 2/3 is larger than 1/2, and asking for two things cannot be easier than asking for one.

Choosing the rule

Three questions sort out almost every problem in this topic.

1. Does the word "not", "never" or "at least" appear? Then the complement rule is probably the short way. 2. Does the question say or? Then add — and ask whether the two events can both happen, because if they can, the overlap must be subtracted once. 3. Does the question say and? Then multiply — and ask whether the first event changes the second, because if it does the events are not independent and the second probability has to be adjusted.

Write the question in symbols before you compute anything. "At least one head in two tosses" becomes 1 - P(tails and tails) = 1 - 1/2 x 1/2 = 3/4, and the arithmetic that follows is easy.

Worked examples

Example 1

Of 20 ferry passengers, 12 carry a lantern, 9 carry a map, and 5 carry both. One passenger is chosen at random. What is P(lantern or map)?

  1. The two events can both happen — five people prove it — so the overlap must be handled.
  2. P(lantern) = 12/20, P(map) = 9/20, P(both) = 5/20.
  3. P(lantern or map) = P(lantern) + P(map) - P(both) = 12/20 + 9/20 - 5/20.
  4. = (12 + 9 - 5)/20 = 16/20 = 4/5.
  5. Check by counting people directly: 7 carry only a lantern, 4 only a map, 5 both, so 16 of the 20 carry at least one. That agrees.

Example 2

A die is rolled twice. What is the probability of at least one six?

  1. "At least one" covers one six or two sixes, so listing cases is slow. Use the complement instead.
  2. The opposite of "at least one six" is "no sixes at all".
  3. The rolls are independent, so P(no six on both) = 5/6 x 5/6 = 25/36.
  4. P(at least one six) = 1 - 25/36 = 11/36.
  5. That is about 0.31 — under a half, which is right: two rolls should not make a six more likely than not.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Aisha reads the marsh signs and reckons the probability of reeds in flower tomorrow is 1/4. What is the probability that it does NOT happen? Give a fraction in lowest terms.

Answer: 3/4

  1. The event and its complement cover everything, so their probabilities add to 1.
  2. P(not) = 1 - 1/4
  3. 1 = 8/8, so P(not) = 8/8 - 2/8 = 6/8
  4. In lowest terms that is 3/4.

Problem 2

Difficulty 3 of 5

A pouch holds 5 blue, 7 silver and 4 green tokens. One token is drawn. What is the probability that it is blue OR silver? Give a fraction in lowest terms.

Answer: 3/4

  1. Total tokens: 5 + 7 + 4 = 16.
  2. P(blue) = 5/16, P(silver) = 7/16.
  3. They cannot both happen, so P(blue or silver) = 5/16 + 7/16 = 12/16.
  4. In lowest terms that is 3/4.

Problem 3

Difficulty 4 of 5

A six-sided die is thrown and then a spinner with 4 equal sections is thrown. The two throws do not affect each other. What is the probability of a number above 4 on the first AND green on the second? Give a fraction in lowest terms.

Answer: 1/12

  1. P(a number above 4) = 2/6 = 1/3
  2. P(green) = 1/4 = 1/4
  3. Independent events joined by "and" multiply: 1/3 x 1/4
  4. = 1/12

Common mistakes

  • Giving P(A) when the question asked for P(not A), or subtracting from the wrong thing instead of from 1.
  • Adding the probabilities of two events joined by "and", which makes the answer bigger when it should be smaller.
  • Forgetting to subtract the overlap in an "or" question, so the people in both groups are counted twice.
  • Adding the denominators as well as the numerators when adding two fractions that already share a denominator.
  • Turning a fraction upside down instead of subtracting it from 1, which can push a "probability" above 1.

What you should be able to do

  • Use P(not A) = 1 - P(A) to find the probability of a complement.
  • Add probabilities for events that cannot both happen, and subtract the overlap when they can.
  • Multiply probabilities for independent events joined by "and".
  • Answer an "at least one" question by working with the complement instead.

Where this fits in the curriculum

Common Core

  • 7.SP.C.7

    Grade 7 — Develop a probability model and use it to find probabilities of events, comparing probabilities from a model with observed frequencies.

  • HSS-CP.A.2

    High school — Understand that two events are independent when the probability of both occurring together is the product of their probabilities.

  • HSS-CP.B.7

    High school — Apply the addition rule P(A or B) = P(A) + P(B) - P(A and B), and interpret it in context.

Ontario

  • G8.D2

    Grade 8 — Describe the likelihood that events will happen, and use that information to make predictions.

    Ontario numbers its Data expectations per grade document and the specific numbering could not be verified line by line here, so this tag names the strand's OVERALL expectation rather than guessing at a specific one.

SAT

Learn these first

This leads on to