REVISION SUMMARY: Probability (NCERT Class 10, Ch 14)
1. Chapter at a glance
- Theoretical (classical) probability is defined only when outcomes are equally likely: \(P(E)=\frac{\text{number of outcomes favourable to }E}{\text{number of all possible outcomes}}\).
- Probability of a sure/certain event is 1; probability of an impossible event is 0.
- For any event \(E\), \(0\leq P(E)\leq1\).
- An elementary event has exactly one outcome; the sum of probabilities of all elementary events of an experiment equals 1.
- Complementary events satisfy \(P(E)+P(\overline{E})=1\), i.e., \(P(\overline{E})=1-P(E)\).
- The definition applies only to experiments with a finite number of equally likely outcomes (Examples 10, 11 and 20 are excluded from examination).
- Well-shuffling, fair coins/dice and random draws ensure equally likely outcomes.
2. Definitions, theorems and results
- Theoretical probability (Laplace): \(P(E)=\frac{\text{number of outcomes favourable to }E}{\text{number of all possible outcomes of the experiment}}\), where outcomes are assumed equally likely.
- Elementary event: an event having only one outcome of the experiment.
- Sure/certain event: an event that is sure to occur; its probability is 1.
- Impossible event: an event that cannot occur; its probability is 0.
- Complementary events: \(E\) and \(\overline{E}\) (“not \(E\)”) satisfy \(P(E)+P(\overline{E})=1\).
- Result (no separate proof required in exam): Sum of probabilities of all elementary events = 1.
- Result: \(0\leq P(E)\leq1\) follows directly from the definition (numerator ≤ denominator).
No theorem in the chapter requires a formal proof in the Board examination; only direct application of the definition and the complementary-event result is expected.
3. Formula sheet
| Formula |
Meaning of symbols |
| \(P(E)=\frac{n(E)}{n(S)}\) |
\(n(E)\)=favourable outcomes, \(n(S)\)=total possible outcomes (equally likely) |
| \(P(\overline{E})=1-P(E)\) |
\(\overline{E}\)=complement of \(E\) |
| \(P(\text{sure event})=1\) |
Event certain to occur |
| \(P(\text{impossible event})=0\) |
Event cannot occur |
| \(0\leq P(E)\leq1\) |
Range of any probability |
| \(\sum P(E_i)=1\) |
Sum over all elementary events \(E_i\) of an experiment |
4. Solved-example patterns
Type A – Single trial with equally likely outcomes (coin, die, ball, card)
Steps: (i) List all possible outcomes and confirm they are equally likely; (ii) count outcomes favourable to the required event; (iii) apply \(P(E)=\frac{\text{favourable}}{\text{total}}\).
Type B – Complementary-event problems (“not E”)
Steps: (i) Find \(P(E)\); (ii) use \(P(\overline{E})=1-P(E)\).
Type C – “At least one” or “exactly one” with two or more identical objects (two coins, two dice)
Steps: (i) Write the sample space as ordered pairs (or list); (ii) count pairs satisfying the condition; (iii) divide by total number of pairs.
Type D – Cards / marbles with colour or face conditions
Steps: (i) Use standard deck facts (52 cards, 4 suits, 4 aces, 12 face cards); (ii) subtract unwanted cards when needed (e.g., not an ace = 52 – 4).
Type E – “Acceptable/reject” or “good/defective” items
Steps: (i) Classify items into required categories; (ii) count favourable items; (iii) divide by total items.
Type F – Finding probability when one event is already known (e.g., two players, two friends)
Steps: (i) Identify the complementary relationship; (ii) subtract from 1.
5. Common mistakes and exam pitfalls
- Forgetting that outcomes must be equally likely before applying the formula.
- Writing total outcomes incorrectly (e.g., counting (1,4) and (4,1) as the same when order matters).
- Using experimental probability instead of theoretical probability.
- Missing the complement relation and calculating “not E” by listing instead of \(1-P(E)\).
- Including outcomes from examples marked “not for examination” (geometric probability).
- Writing probability >1 or <0 (sign error or wrong counting).
- Confusing “face card” with “ace” or forgetting there are only 12 face cards in a deck.