Listing outcomes for one and two events

Maths · Listing outcomes

Roll a dice, then spin a spinner

A fair six-sided dice is rolled, then a spinner with sectors A, B and C is spun. Tap a dice number, then a letter, to walk one path. Where the path ends is one possible outcome.

Dice → Spinner

On Roll the dice, then spin. 6 branches to choose from.

Fix one dice number, pair it with every letter, then move on to the next number. Six dice numbers, three letters each: the tree ends 18 times, so there are 18 outcomes, with nothing missed and nothing doubled.

Maths · Likelihood

Equally likely, or not?

Each trial below has outcomes. Decide whether its outcomes are equally likely, then read why.

7 of 7 still to sort.

Predict, then check

No counting yet. Make a call, then see what the lists say.

A regular six-sided dice is rolled. Which event is more likely: 'a multiple of 3' or 'not a multiple of 3'?

What you need to know

• A trial is one predefined test, like spinning a spinner once. An outcome is just one result of it, not all the results.
• The sample space is all the possible outcomes. Write it after ξ in curly brackets, like ξ = {win, lose}. Their order doesn't matter.
• List each distinct outcome once. A spinner with two win sectors and two lose sectors still has only two outcomes.
• Have a goMaya's spinner has 3 red sectors and 1 blue sector. She says, 'Four sectors, so four outcomes.' How many outcomes does it really have?

  Two: red and blue.

  A sample space lists each distinct outcome once, so three red sectors are still just the one outcome, red.

• What you call the trial decides the sample space: one dice roll gives 1 to 6, but a whole game gives its possible winners.
• Outcomes are equally likely when nothing favours one, like same-sized sectors. A larger sector, or extra counters showing one letter, breaks that.
• An event is a subset of the sample space. If outcomes are equally likely, the event with more outcomes is more likely.
• Have a goA regular six-sided dice is rolled. Write the event 'an even number' as a list of outcomes.

  {2, 4, 6}

  An event is a subset of the sample space, so it holds only the outcomes that fit and leaves out the rest.

• For two events, fix one outcome of the first and pair it with every outcome of the second. Then move on.
• Have a goSpin a win/lose spinner, then flip a coin (heads or tails). Fix 'win' on the first spin. Which outcomes start with win?

  win, heads and win, tails.

  Fixing one outcome of the first event means pairing it with every outcome of the second before you move on.

• In a two-stage trial, order matters: scissors–paper and paper–scissors use the same choices but are different outcomes.
• Count the ends of a systematic list, or multiply: six dice outcomes and three spinner outcomes give 18 outcomes.
• Outcomes can sit in both lists, either, one only, or neither. An event with no outcomes is ∅, not 0.

The big picture

A trial is one test and an outcome is one result of it. The sample space, written ξ = {…}, lists each distinct outcome once. For two events, fix one outcome of the first and pair it with every outcome of the second before moving on: the ends of the tree are your list, and the count is the two counts multiplied. Order matters in a two-stage trial, an event is a subset of the sample space, and an event with no outcomes is ∅.

⚠ Watch out

Calling scissors–paper and paper–scissors the same outcome. They use the same two choices, but they are different outcomes, because it matters which person chose which. Also watch the other slip: a list with the right number of entries can still repeat one outcome and miss another.

Memory hook

Fix one, pair it with them all, then move on. Count the ends and you've counted the outcomes.

Check yourself

A spinner shows red, blue or green, then a coin is flipped. Fix the spinner first and list every outcome. How many are there, and why?