Counting Outcomes — the Counting Principle (Grade 7)
Before you can find a probability by counting, you need the sample space: every outcome that can happen.
Understanding counting outcomes — the counting principle
When an experiment has stages, such as flipping a coin and then rolling a die, multiply the number of outcomes at each stage to count them all (2 × 6 = 12); this is the counting principle. Then list the outcomes in a table or tree diagram, so you can pick out exactly the ones that make up the event you care about. Counting works for probability only when every listed outcome is equally likely.
Seeing it in action
Worked example
Flip a coin and roll a die. How many outcomes are there, and what is P(heads and an even number)?
Outcomes: 2 × 6 = 12, from H1, H2, … H6 to T1, T2, … T6.
Heads and even: H2, H4 and H6, so 3 of the 12 outcomes. → P = 3/12 = 1/4.
| Coin | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| H | H1 | H2 | H3 | H4 | H5 | H6 |
| T | T1 | T2 | T3 | T4 | T5 | T6 |
Heads and an even number: H2, H4 and H6, highlighted. That is 3 of 12 equally likely outcomes, so P = 3/12 = 1/4.
Worked example 2: two dice
Roll two dice. What is P(a total of 7)?
Outcomes: 6 × 6 = 36. A total of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), so 6 outcomes. → P = 6/36 = 1/6.
A common mistake, and the fix
Counting (2, 5) and (5, 2) as one outcome.
A 2 on the first die and a 5 on the second is a different outcome from a 5 then a 2, so a total of 7 happens 6 ways out of 36, not 3 ways out of 21. A 6 by 6 table keeps every ordered pair separate.
Try a few
Flip two coins: how many outcomes, and P(both heads)?
HH, HT, TH, TT; only HH.
Spin a spinner with three equal sectors (red, blue and green) and flip a fair coin: P(red and tails)?
3 × 2 = 6 equally likely outcomes; only red-tails.
Roll two dice: P(a total of 2)?
only (1, 1) of the 36.
Spinner
A calm probability game for reading spinner sectors, fractions, complements, and comparisons.
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