Probability begins with a complete universe
The sample space is the complete set of possible outcomes. An event is a subset selected by a condition; missing or duplicated sample outcomes corrupt every later probability.
Learning goals
- Distinguish experiments, outcomes, events and sample spaces.
- List and count sample spaces for common random experiments.
- Translate verbal event conditions into favourable outcome sets.
Concept lesson
Systematic finite-set listing
Use an ordered list, table or tree. Complete every branch once and compare the list count with the multiplication rule.
Number-property vocabulary
Review odd, even, prime, square, cube, multiple, divisible by both, at least, at most, greater than and not greater than.
Experiment, outcome, sample space and event
A random experiment has known possible outcomes but an uncertain trial result. One realised result is an outcome; all possible outcomes form S; outcomes satisfying a stated condition form event E.
Ordered outcomes for repeated trials
For two coins use HH, HT, TH, TT. For two dice use ordered pairs (first, second), giving 6 times 6 outcomes; swapping coordinates generally gives a different elementary outcome.
Two-dice sum table
Build a 6-by-6 ordered-pair grid. For sum 5 select (1,4),(2,3),(3,2),(4,1); retain swapped pairs as distinct outcomes.
Translate a card-number condition
For cards 5 through 30, write the sample range explicitly, then list multiples, squares or intersections from that range. Interpret both as intersection and or as union before counting.
Tree-list-count triangle
Represent two-stage coin, colour or spinner trials by a tree and an ordered list, then verify the same count using multiplication.
Ordered-pair grid audit
Complete rows by first die and columns by second die; check 36 unique pairs and test event conditions on pair values or their sums.
Condition-to-set translator
Underline comparison and logical words, rewrite them symbolically, list favourable outcomes from S and check every listed outcome satisfies the original sentence.
Select the representation
Choose a list for one-stage finite outcomes, tree for repeated small trials, grid for two dice and filter table for number properties; justify completeness and uniqueness.
Sample-space answer protocol
State S or its count, state the favourable subset or its count, and show the condition filter. For repeated trials preserve order explicitly.
Sample-space mastery
Construct coin and dice sample spaces, translate compound number conditions and verify every list is exhaustive and duplicate-free.