Complementary events

Activities to try

1. 1

   Unit bar: complement and the overlap trap

   Why it works: Non-overlapping segments of a complete outcome set fill the bar to exactly 1, and P(not A) is simply the rest. Adding probabilities only works when the events share no outcome; overlapping events double-count the shared outcome, so the sum overstates the true probability — here it reaches exactly 1 while the true value is only 5/6.

   1. Use a bar representing probability 1, divided into the six equally likely outcomes of a fair die (each 1/6).
   2. Choose an event, e.g. 'rolling a 6'. Shade that segment, then see that P(not 6) is the whole bar minus that one segment: 1 − 1/6 = 5/6. Confirm this matches the five remaining faces without counting them one by one.
   3. Switch to overlap mode and select BOTH 'even' (2, 4, 6) and 'prime' (2, 3, 5). Shade each region on the bar.
   4. Add P(even) = 1/2 and P(prime) = 1/2 to get 1, then compare with the bar: the arithmetic sum of the two lengths reaches the whole bar (suggesting 'certain'), yet the actual shaded union only covers five of the six segments (5/6). The extra 1/6 is the number 2, which sits in both regions and has been counted twice.
   5. List the distinct outcomes that are even OR prime: 2, 3, 4, 5, 6 — five faces — so P(even or prime) = 5/6. Note that 1 is neither, so the event is NOT certain.
2. 2

   Predict, roll, explain: is 'even or prime' certain?

   Why it works: An outcome (1) that is neither even nor prime disproves the 'certain' claim and exposes that adding overlapping probabilities over-counts the shared outcome. Checking for a shared outcome BEFORE adding is the deciding skill.

   1. Before rolling: 'On a fair die, is rolling a number that is even OR prime certain? Predict yes or no, give a probability, and explain your reasoning.'
   2. Record the prediction (a learner who adds 1/2 + 1/2 will predict 'certain, probability 1').
   3. Roll the die several times. A 1 is likely to appear but is not guaranteed in a small number of rolls (each roll has a 1/6 chance). If a 1 appears, note that it is neither even nor prime, which contradicts a 'certain' prediction. If no 1 appears, check the faces anyway: 1 is a possible outcome that is neither even nor prime, so the event still cannot be certain. Not rolling a 1 does not show the event is certain.
   4. Revisit the reasoning: list the even-or-prime outcomes, find the double-counted 2, and correct the probability to 5/6.
   5. Discuss: when is it safe to add two probabilities, and when is it not?
3. 3

   Sort and check: mutually exclusive or overlapping, and valid distributions

   Why it works: Testing for overlap is a prerequisite to adding; sum-to-1 only applies to a complete (exhaustive) set of mutually exclusive outcomes; complements are found by subtracting from 1.

   1. Sort pairs of events into 'mutually exclusive' (cannot both happen) or 'overlapping' (share at least one outcome). Include die examples such as {even}/{prime} (overlap at 2), {1,2,3}/{4,5,6} (no overlap), {multiple of 3}/{even} (overlap at 6).
   2. For each overlapping pair, list the shared outcome(s) explicitly.
   3. Fill in complements directly: P(not 6) = 1 − 1/6 = 5/6; P(not even) = 1 − 1/2 = 1/2.
   4. Examine several probability distributions and decide which are invalid because the probabilities do not sum to 1, or because the list is not exhaustive. Explain each decision.
   5. Verify that the probabilities listed for a given spinner sum to 1, and find any missing probability by subtracting from 1.