Probability
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Conditional Probability
Part of: Conditional Probability and Independence
Knowing F occurred shrinks the sample space to F, giving P(E|F) = P(E∩F)/P(F). Whole-space and self-conditioning equal one, complements subtract from one, and conditioning never works on zero-probability events.
Why it matters. Tests, surveys and games all update chances on fresh information.
Formulas
- P(E|F) is P(E∩F) over P(F)
- Whole space conditions to one
- Complements subtract from one
Key words
- Conditional probability
- : Chance of E with F as the sample space.
- Reduced sample space
- : The known event acting as new total.
- Complement rule
- : P(E′|F) is one minus P(E|F).
- Zero-probability guard
- : Conditioning needs P(F) nonzero.
Try it in two minutes
- List the reduced sample space first.
- Count favourable inside the reduction.
- Divide by the reduced total.
Try at home. Find P(even | above 4) on a fair die.
Multiplication and Independence
Part of: Conditional Probability and Independence
Chained stages multiply: P(A∩B) = P(A)·P(B|A), with shrinking totals without replacement and steady totals with it. Independent stages ignore each other, splitting the product into P(A)·P(B).
Why it matters. Draws, throws and surveys chain exactly this way.
Formulas
- Chain stages by multiplying
- Independent means product splits
- Kept-aside draws shrink totals
Key words
- Multiplication rule
- : Chaining as P(A) times P(B|A).
- Independent events
- : Events ignoring each other with product intersections.
- Without replacement
- : Draws shrinking later totals.
- With replacement
- : Draws keeping totals steady.
Try it in two minutes
- Name the stages in order.
- Update totals after each draw.
- Multiply stage chances.
Try at home. Draw two cards with and without replacement; compare both-red chances.
Bayes Theorem
Part of: Bayes and Repeated Trials
Total probability adds each cause times its evidence rate, and Bayes reverses the question into cause-given-evidence. Partitions must be disjoint and covering, priors weight each cause, and the denominator always totals the evidence.
Why it matters. Diagnoses, filters and inspections all reverse evidence into causes.
Formulas
- Total evidence sums cause times rate
- Bayes divides target cause by total
- Partitions stay disjoint and covering
Key words
- Partition
- : Disjoint events covering the sample space.
- Prior probability
- : A cause chance before evidence.
- Total probability
- : Evidence summed over all causes.
- Posterior probability
- : A cause chance after evidence.
Try it in two minutes
- List causes with priors.
- Weigh each cause by its evidence rate.
- Divide the target cause by the total.
Try at home. Two coins in a pocket, one double-headed: flip heads, find P(double-headed).
Repeated Trials
Part of: Bayes and Repeated Trials
Repeated fair trials multiply single chances, complements catch at-least-one events, truth-tellers blend honesty with luck through Bayes, and alternate-turn wins sum geometric series over odd or even trials.
Why it matters. Games, tests and contests all repeat chances this way.
Formulas
- Repeat fair trials by multiplying
- At-least-one is one minus none
- Contests sum over winning trials
Key words
- Independent trials
- : Repeats ignoring earlier outcomes.
- At-least-one rule
- : One minus the none chance.
- Truth-teller blend
- : Honesty times luck over all reports.
- Alternate-turn win
- : First success across swapping players.
Try it in two minutes
- Check trials are fair and independent.
- Multiply for all-runs, complement for at-least-one.
- Sum over winning trial numbers for contests.
Try at home. Toss a coin 3 times and list all 8 outcomes with chances.
What your child will practise
- Bayes Theorem: Candy Quick Fire (Quick Fire)
- Bayes Theorem: Space Puzzle (Clue Puzzle)
- Conditional Probability: Ocean Puzzle (Clue Puzzle)
- Multiplication and Independence: Jungle Quest (Flip and Match)
- Repeated Trials: Ocean Quick Fire (Quick Fire)
- Quiz: Bayes Theorem (8 questions)
- Quiz: Conditional Probability (8 questions)
- Quiz: Multiplication and Independence (8 questions)
- Quiz: Repeated Trials (8 questions)