Probability is a part-to-whole ratio
For equally likely outcomes, probability compares favourable outcomes with the complete sample space. A subset cannot contain fewer than zero or more than all outcomes, so probability lies from 0 to 1.
Learning goals
- Calculate theoretical probability from equally likely outcomes.
- Use probability bounds and classify event certainty.
- Solve contextual and inverse probability problems.
Concept lesson
Sample set and event set
Write or count the complete sample space S and the favourable event set E before forming a probability ratio.
Equivalent probability forms
Convert fractions, decimals and percentages consistently; 15 percent equals 0.15 and lies on the probability scale.
Theoretical probability formula
When elementary outcomes are equally likely, P(E)=n(E)/n(S). Use one fixed denominator n(S) for all events from the same experiment and simplify only after counting.
Probability scale and certainty
P=0 means impossible, P=1 sure, and values strictly between mean possible but uncertain. No event probability can be negative or greater than one.
Probability from a die or ordered pair grid
List favourable die faces or ordered dice pairs, retain the full sample count of 6 or 36 as denominator, and simplify the ratio.
Bag and frequency-table probability
Add all item or student counts for the denominator. Add counts in rows satisfying the event for the numerator; do not use the number of colours or table categories.
Unknown category count
If there are 6 blue and x red objects, total is 6+x. Set the given blue probability equal to 6/(6+x), solve for x, require a non-negative integer and verify both probabilities.
Numerator-denominator ledger
Write the experiment, n(S), event, n(E), ratio and simplified value in separate rows; explain why every denominator outcome is equally likely.
Probability-scale validator
Place fractions, decimals and percentages on 0-to-1 scale; reject invalid candidates and classify endpoints as impossible or sure.
Count items, not labels
From narrative or tabular data, mark the selectable elementary items, total their frequencies, then total favourable items across all qualifying categories.
Inverse contextual probability
Translate a bag or population statement into favourable and total expressions containing an unknown, solve the probability equation, require a feasible count and reconstruct the complete distribution.
Theoretical-probability answer protocol
State n(S), state or list n(E), write P(E), simplify and test 0<=P<=1. For inverse items, verify an admissible item count.
Theoretical-probability mastery
Calculate probabilities from a sample space and a frequency table, reject invalid values and solve an unknown-count problem with verification.