ICSE Class 10
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๐Ÿ“– Summaries โ€บ Mathematics

Probability

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Chapter in a nutshell: Probability measures how likely an event is, as a number from 0 (impossible) to 1 (certain). For equally likely outcomes, $P(E)=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$. The probability of "not E" is $1-P(E)$.

1. Key Ideas & Terms

  • Experiment / trial: an action with uncertain results (toss a coin, roll a die).
  • Sample space (S): the set of all possible outcomes; $n(S)$ = its size.
  • Event (E): a set of favourable outcomes; $n(E)$ = its size.
  • Equally likely: each outcome has the same chance (fair coin/die/cards).

2. Formulas (quick reference โ€” all of them)

  • Probability of an event: $P(E)=\dfrac{n(E)}{n(S)}=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$
  • Range: $0\le P(E)\le 1$.
  • Sure (certain) event: $P=1$; impossible event: $P=0$.
  • Complement: $P(\text{not }E)=1-P(E)$; so $P(E)+P(\bar E)=1$.
  • Sum of all outcome probabilities $=1$.

3. Prerequisite Facts / Sample Spaces (in full)

  • Coin: outcomes {H, T}; two coins โ†’ {HH, HT, TH, TT} (4 outcomes).
  • Die: {1, 2, 3, 4, 5, 6} (6 outcomes); two dice โ†’ 36 outcomes.
  • Pack of 52 cards: 4 suits (โ™  โ™ฅ โ™ฆ โ™ฃ); 26 red + 26 black; 13 of each suit; face cards = J, Q, K (12 total); 4 aces; 4 of each rank.
  • Fractions/percentages to express the probability; reduce to lowest terms.

4. Worked Example (one, for the method)

Q. A card is drawn from a well-shuffled pack of 52. Find P(it is a king). Solution: $n(E)=4$ kings, $n(S)=52$. $P=\dfrac{4}{52}=\dfrac{1}{13}$.

5. Common Mistakes to Avoid

  • Giving a probability greater than 1 or negative (must be between 0 and 1).
  • Wrong sample-space size (e.g. two dice have 36, not 12, outcomes).
  • Forgetting to reduce the fraction.
  • Miscounting cards (face cards = 12; ace is not a face card).
  • Using $P(E)$ when $P(\text{not }E)=1-P(E)$ is asked.

6. Likely Exam Questions (with crisp answers)

  1. Write the probability formula. โ†’ $P(E)=\frac{\text{favourable}}{\text{total}}$.
  2. Range of probability? โ†’ 0 to 1.
  3. Probability of a sure event? โ†’ 1.
  4. Probability of an impossible event? โ†’ 0.
  5. State the complement rule. โ†’ $P(\text{not }E)=1-P(E)$.
  6. Number of outcomes when two dice are rolled? โ†’ 36.
  7. P(getting a head) for a fair coin? โ†’ 1/2.
  8. How many face cards in a pack? โ†’ 12.
  9. P(drawing a red card) from 52? โ†’ 26/52 = 1/2.
  10. P(getting a number > 4 on a die)? โ†’ 2/6 = 1/3.
  11. Sum of probabilities of all outcomes? โ†’ 1.
  12. P(an ace) from a pack? โ†’ 4/52 = 1/13.

Extended & Prerequisite Formulas (full)

  • $P(E)=\tfrac{n(E)}{n(S)}$; 0โ‰คPโ‰ค1; P(sure)=1, P(impossible)=0; P(not E)=1โˆ’P(E); ฮฃP=1.
  • Sample spaces: coin {H,T}; 2 coins 4; die 6; 2 dice 36; pack 52 (26 red+26 black, 13/suit, 12 face cards, 4 aces).
  • Prereq: fractions in lowest terms; favourable vs total outcomes; equally-likely assumption.