A sample space is the full list of outcomes, and a systematic list fixes the order in which you write them. Hold the first item steady while the second changes through all its values, then move to the next first item.
This lesson opens the combined-event probability section. Once your lists are reliable, the next skill is to distinguish independent and dependent events.
How do I list outcomes without missing any?
Fix one item and vary the other. Write H with 1, 2, 3, 4, 5, 6 in turn, then T with 1, 2, 3, 4, 5, 6.
Before you read off any probability, check the total by multiplying: 2 outcomes for the coin times 6 for the die is 12.
Worked example: a coin and a die
A fair coin is tossed and a fair die is rolled. Find the probability of a head and an even number.
Sample space: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6. That is 12 outcomes, and every one is equally likely.
Favourable outcomes: H2, H4, H6, which is 3 outcomes.
P(head and even) = 3/12 = 1/4.
Worked example: two dice as a grid
For two dice, a 6 by 6 grid lists each pair once. The row is the first die and the column is the second die.
A sum of 7 comes from (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1). That is 6 of 36 outcomes, so P(sum 7) = 6/36 = 1/6.
A sum greater than 9 comes from (4, 6), (5, 5), (6, 4), (5, 6), (6, 5) and (6, 6). That is also 6 of 36, so P(sum > 9) = 1/6.
The mistake that merges outcomes
A common slip is to list the outcomes of two coins as HH, HT and TT, treating HT and TH as one outcome. That gives P(two heads) = 1/3.
The two coins are different objects, so a head on the first coin and a tail on the second is a different outcome from the reverse. The list HH, HT, TH, TT has 2 × 2 = 4 entries, so P(two heads) = 1/4. The count check catches the error, because 3 is not 2 × 2.
Check yourself
A spinner has three equal sectors numbered 1, 2 and 3, and a coin is tossed. List the sample space and find P(tail and an odd number).
Answer
Fix the spinner and vary the coin: H1, T1, H2, T2, H3, T3. That is 3 × 2 = 6 outcomes.
The outcomes with a tail and an odd number are T1 and T3.
P = 2/6 = 1/3.
What to study next
Once you can list outcomes reliably, see whether the second stage changes after the first in distinguishing independent and dependent events. Test your lists later with the combined-event probability practice set.
If you want a teacher to watch how you list outcomes on your own questions, see online one-to-one Mathematics tuition.