ICSE Class 10 Mathematics • Chapter 15
| Term | Definition | Example |
|---|---|---|
| Experiment | Activity with well-defined outcomes | Tossing a coin, rolling a die |
| Trial | Single performance of experiment | One coin toss |
| Outcome | Possible result of a trial | Getting Head, Getting 6 |
| Sample Space (S) | Set of all possible outcomes | $S = \{H, T\}$ for coin |
| Event (E) | A subset of sample space | "Getting even number" |
| Favourable Outcomes | Outcomes satisfying the event | {2, 4, 6} for even on die |
$P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$
Important Properties:
| Experiment | Sample Space | n(S) |
|---|---|---|
| 1 Coin | $\{H, T\}$ | 2 |
| 2 Coins | $\{HH, HT, TH, TT\}$ | 4 |
| 3 Coins | $\{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$ | 8 |
| n Coins | - | $2^n$ |
| Experiment | Sample Space | n(S) |
|---|---|---|
| 1 Die | $\{1, 2, 3, 4, 5, 6\}$ | 6 |
| 2 Dice | $(1,1), (1,2), ..., (6,6)$ | 36 |
| n Dice | - | $6^n$ |
Common Events for 2 Dice (36 outcomes):
| Event | Outcomes | Count | P(E) |
|---|---|---|---|
| Doublet (same on both) | (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) | 6 | $\frac{1}{6}$ |
| Sum = 7 | (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) | 6 | $\frac{1}{6}$ |
| Sum = 2 | (1,1) | 1 | $\frac{1}{36}$ |
| Sum = 12 | (6,6) | 1 | $\frac{1}{36}$ |
| Sum > 10 | (5,6),(6,5),(6,6) | 3 | $\frac{1}{12}$ |
Complete Deck Structure:
| Category | Details | Count |
|---|---|---|
| Total Cards | - | 52 |
| Suits | Hearts ♥, Diamonds ♦, Spades ♠, Clubs ♣ | 4 suits × 13 = 52 |
| Red Cards | Hearts ♥ + Diamonds ♦ | 26 |
| Black Cards | Spades ♠ + Clubs ♣ | 26 |
| Face Cards | Jack, Queen, King (each suit has 3) | 12 |
| Number Cards | 2 to 10 (each suit has 9) | 36 |
| Aces | Ace of each suit | 4 |
| Each Rank | e.g., all Kings, all 7s | 4 |
| Event | Favourable | P(E) |
|---|---|---|
| A King | 4 | $\frac{4}{52} = \frac{1}{13}$ |
| A Red Card | 26 | $\frac{26}{52} = \frac{1}{2}$ |
| A Face Card | 12 | $\frac{12}{52} = \frac{3}{13}$ |
| A Spade | 13 | $\frac{13}{52} = \frac{1}{4}$ |
| An Ace or King | 8 | $\frac{8}{52} = \frac{2}{13}$ |
| A Red King | 2 | $\frac{2}{52} = \frac{1}{26}$ |
| Not an Ace | 48 | $\frac{48}{52} = \frac{12}{13}$ |
Complementary Event (E'): Event that E does NOT occur. $P(E') = 1 - P(E)$
Simple Event: Event with only one outcome.
Compound Event: Event with more than one outcome.
Shortcut for "At Least One":
$P(\text{at least one}) = 1 - P(\text{none})$
This is much easier than calculating all "at least one" cases!
Example 1: A bag contains 5 red, 4 blue, and 3 green balls. A ball is drawn at random. Find probability of:
(a) Red ball (b) Not green (c) Blue or green
Solution:
Total balls = 5 + 4 + 3 = 12
(a) P(Red) = $\frac{5}{12}$
(b) P(Not green) = $1 - P(\text{green}) = 1 - \frac{3}{12} = \frac{9}{12} = \frac{3}{4}$
(c) P(Blue or Green) = $\frac{4+3}{12} = \frac{7}{12}$
Example 2: Two dice are thrown. Find probability of:
(a) Sum = 9 (b) Sum is a prime (c) Doublet of even number
Solution:
n(S) = 36
(a) Sum = 9: (3,6), (4,5), (5,4), (6,3) → P = $\frac{4}{36} = \frac{1}{9}$
(b) Prime sums: 2,3,5,7,11
(c) Even doublets: (2,2), (4,4), (6,6) → P = $\frac{3}{36} = \frac{1}{12}$
Example 3: Cards marked 1 to 20 are placed in a box. Find P(divisible by 3 or 5).
Solution:
n(S) = 20
Divisible by 3: {3, 6, 9, 12, 15, 18} = 6 cards
Divisible by 5: {5, 10, 15, 20} = 4 cards
Divisible by both (15): 1 card (counted twice)
Divisible by 3 OR 5 = 6 + 4 − 1 = 9
P = $\frac{9}{20}$
| Concept | Formula/Value |
|---|---|
| Probability | $\frac{n(E)}{n(S)}$ |
| Range | $0 \leq P(E) \leq 1$ |
| Complementary | $P(E') = 1 - P(E)$ |
| n coins | n(S) = $2^n$ |
| n dice | n(S) = $6^n$ |
| Deck of cards | n(S) = 52 |
BOARD From a pack of 52 cards, a card is drawn at random. Find probability that the card is: (a) a face card of red colour (b) neither a spade nor a king
BOARD Two dice are thrown simultaneously. Find probability that: (a) sum is 10 (b) sum is at least 10 (c) a doublet of prime numbers
HOTS A bag contains tickets numbered 11, 12, 13, ..., 30. A ticket is drawn at random. Find probability that the number is: (a) multiple of 7 (b) greater than 20 and even