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The chance that something does not happen

Why the chance of "not" is 1 minus the chance of "yes", and how it turns "at least one six" into an easy question.

If the weather forecast says there is a 30% chance of rain, then there is a 70% chance that it stays dry. You know this without any calculation, because something either happens or it does not. This simple idea is one of the most useful tools in probability. It turns some very hard questions, like "what is the chance of at least one six in two rolls?", into easy ones.

Yes and no add up to 1

Every time, an event either happens or it does not happen. There is no third choice. So the chance of "yes" and the chance of "no" together make the whole, which is 1 (or 100%). The event "it does not happen" is called the complement of the event.

Remember

P(not A) = 1 − P(A)

P(A) means "the probability of A". With percents, use 100% instead of 1.

You can always find a "not" probability in two ways. You can count the outcomes that do NOT work. Or you can find the chance that the event happens and take it away from 1. Both give the same answer, so you can use one to check the other.

Not this color, not this card

Example 1

A bag has 2 red, 5 blue and 3 green marbles. You take one without looking. What is the probability that it is NOT blue?

  1. All the marbles: 2 + 5 + 3 = 10.
  2. Way 1, count: the marbles that are not blue are 2 + 3 = 5, so 5/10.
  3. Way 2, take away: blue is 5/10, so not blue is 1 − 5/10 = 5/10.
  4. Simplify: 5/10 = 1/2.

Answer: 1/2

To take a fraction away from 1, write 1 as a fraction with the same bottom number. So 1 = 8/8, and 1 − 3/8 = 8/8 − 3/8 = 5/8.

Example 2

This spinner has 8 equal parts: 3 red, 3 yellow and 2 blue. What is the probability that the arrow does NOT stop on blue?

  1. Blue: 2 of the 8 parts, so 2/8.
  2. Not blue: 1 − 2/8 = 6/8.
  3. Simplify: 6/8 = 3/4.

Answer: 3/4

Example 3

You take one card from a deck of 52 cards (4 suits of 13). What is the probability that it is NOT a club?

  1. There are 13 clubs, so 52 − 13 = 39 cards are not clubs.
  2. 39/52. Divide the top and the bottom by 13: 3/4.
  3. Check: a club is 1/4, and 1 − 1/4 = 3/4.

Answer: 3/4

With percents

Percents work the same way, with 100% as the whole. If a team wins 60% of its games, then it does not win 100% − 60% = 40% of them. If a shop says 8% of its bulbs break in the first year, then 92% do not.

Example 4

The chance that a train is late is 15%. What is the chance that it is not late?

  1. Late and not late together make 100%.
  2. 100% − 15% = 85%.

Answer: 85%

At least one: count the opposite

At least one means one or more. "At least one 6 in two rolls" is true if the first roll is a 6, or the second, or both. Counting all those cases is slow and easy to get wrong. The opposite is much simpler: the opposite of "at least one 6" is "no 6 at all".

  1. Find the chance that it never happens.
  2. Take that away from 1.

To find the chance that two things both happen, when one cannot change the other (like two rolls of a die), multiply their chances. (The lesson on two things at once explains why.) A roll that is not a 6 has a chance of 5/6, so two rolls with no 6 have a chance of 5/6 × 5/6.

Example 5

You roll a die twice. What is the probability of getting at least one 6?

  1. No 6 on one roll: 5/6.
  2. No 6 on both rolls: 5/6 × 5/6 = 25/36.
  3. At least one 6: 1 − 25/36 = 36/36 − 25/36 = 11/36.

Answer: 11/36

You can check this by counting. Two rolls give 36 results. The first roll is a 6 in 6 of them, and the second roll is a 6 in 6 of them. But the result 6 and 6 is in both lists, so count it once: 6 + 6 − 1 = 11 results.

Exactly one is different from at least one. "Exactly one 6" means one roll is a 6 and the other is not. A 6 and then not a 6 is 1/6 × 5/6 = 5/36. Not a 6 and then a 6 is also 5/36. Together: 10/36 = 5/18. The three answers fit together: exactly one 6 (10/36) plus two 6s (1/36) is at least one 6 (11/36).

Example 6

You flip three coins. What is the probability that at least one shows heads?

  1. The opposite of at least one head is no heads: tails, tails, tails.
  2. Three tails: 1/2 × 1/2 × 1/2 = 1/8.
  3. At least one head: 1 − 1/8 = 7/8.

Answer: 7/8

Try it yourself

  1. 1.A spinner has 5 equal parts: 2 green, 2 blue and 1 red. What is the probability that the arrow does NOT stop on green?

    Show the answer

    3/5

    Green is 2/5, so not green is 1 − 2/5 = 3/5.

  2. 2.A bag has 4 red, 1 yellow and 7 purple marbles. What is the probability that a marble taken without looking is NOT red?

    Show the answer

    2/3

    8 of the 12 marbles are not red: 8/12 = 2/3.

  3. 3.There is a 12% chance that a flight is cancelled. What is the chance that it is not cancelled?

    Show the answer

    88%

    100% − 12% = 88%.

  4. 4.You take one card from a deck of 52. What is the probability that it is NOT an ace?

    Show the answer

    12/13

    An ace is 4/52 = 1/13, so not an ace is 1 − 1/13 = 12/13.

  5. 5.You flip a coin 5 times. What is the probability of at least one tail?

    Show the answer

    31/32

    No tails means five heads: 1/2 multiplied by itself 5 times, which is 1/32. So at least one tail is 1 − 1/32 = 31/32.

  6. 6.You roll a die three times. What is the probability of at least one 6?

    Show the answer

    91/216

    No 6 at all is 5/6 × 5/6 × 5/6 = 125/216. So at least one 6 is 1 − 125/216 = 91/216.

Ready to test yourself?

Play Probability: 20 seconds per question, and it gets harder as you go.