Probability
Introduction
Probability measures the likelihood of an event occurring. In ICSE Class 10, you learn to compute probabilities for simple random experiments involving coins, dice, playing cards, and real-life scenarios.
Key Terms
- Random experiment — An experiment whose outcome cannot be predicted in advance (e.g., tossing a coin, rolling a die).
- Sample space (S) — The set of all possible outcomes of a random experiment.
- Event (E) — A subset of the sample space (a specific outcome or set of outcomes).
- Probability — The measure of how likely an event is to occur.
Formula
P(E) = Number of favourable outcomes / Total number of possible outcomes
P(E) = n(E) / n(S)
Properties of Probability
- 0 ≤ P(E) ≤ 1
- P(E) + P(not E) = 1
- P(impossible event) = 0
- P(certain event) = 1
Common Sample Spaces
| Experiment | Sample Space | n(S) |
|---|---|---|
| Tossing one coin | {H, T} | 2 |
| Tossing two coins | {HH, HT, TH, TT} | 4 |
| Rolling one die | {1, 2, 3, 4, 5, 6} | 6 |
| Rolling two dice | 36 ordered pairs | 36 |
| Drawing a card from 52 | 52 distinct cards | 52 |
Card-Based Problems (Standard Deck of 52)
A standard deck has:
- 4 suits: Spades ♠, Hearts ♥, Clubs ♣, Diamonds ♦
- 13 cards per suit: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K
- Face cards: J, Q, K (12 face cards total, 3 per suit)
- Red suits: Hearts, Diamonds (26 red cards)
- Black suits: Spades, Clubs (26 black cards)
- Aces: 4 (one per suit)
Worked Examples
Example 1: Coin Toss
Two unbiased coins are tossed. Find the probability of getting (a) two heads, (b) at least one head, (c) exactly one tail.
Solution: Sample space S = {HH, HT, TH, TT}, n(S) = 4
(a) E = {HH}, n(E) = 1 P(two heads) = ¹/₄
(b) E = {HH, HT, TH}, n(E) = 3 P(at least one head) = ³/₄
(c) E = {HT, TH}, n(E) = 2 P(exactly one tail) = ²/₄ = ¹/₂
Example 2: Dice Problem
A die is rolled once. Find the probability of getting (a) an even number, (b) a number greater than 4, (c) a prime number.
Solution: S = {1, 2, 3, 4, 5, 6}, n(S) = 6
(a) Even numbers: {2, 4, 6}, n(E) = 3 P(even) = ³/₆ = ¹/₂
(b) Numbers > 4: {5, 6}, n(E) = 2 P(> 4) = ²/₆ = ¹/₃
(c) Prime numbers: {2, 3, 5}, n(E) = 3 P(prime) = ³/₆ = ¹/₂
Example 3: Card Problem
A card is drawn from a well-shuffled pack of 52 cards. Find the probability that it is (a) an ace, (b) a red queen, (c) a face card, (d) a spade.
Solution: n(S) = 52
(a) Aces: 4, P(ace) = ⁴/₅₂ = ¹/₁₃
(b) Red queens: Queen of Hearts + Queen of Diamonds = 2 P(red queen) = ²/₅₂ = ¹/₂₆
(c) Face cards: 12 (3 per suit × 4 suits) P(face card) = ¹²/₅₂ = ³/₁₃
(d) Spades: 13 P(spade) = ¹³/₅₂ = ¹/₄
Example 4: Two Dice
Two dice are rolled. Find the probability that the sum of the numbers is (a) 7, (b) at least 10.
Solution: n(S) = 36
(a) Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes P(sum = 7) = ⁶/₃₆ = ¹/₆
(b) Sum ≥ 10: (4,6), (5,5), (5,6), (6,4), (6,5), (6,6) — 6 outcomes P(sum ≥ 10) = ⁶/₃₆ = ¹/₆
Example 5: Birthday Problem (Non-Leap Year)
Find the probability that a randomly selected person has a birthday in (a) January, (b) the month starting with J.
Solution: (a) January has 31 days, year has 365 days P(January) = ³¹/₃₆₅
(b) Months starting with J: January (31), June (30), July (31) = 92 days P(J-month) = ⁹²/₃₆₅
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Counting (H, T) and (T, H) as the same outcome in two-coin toss | They are distinct outcomes — order matters |
| Including Joker cards in a deck of 52 | Standard deck of 52 has NO jokers |
| Writing P(E) > 1 | Probability is always between 0 and 1 |
| Reducing fractions incorrectly | Always express probability in simplest form |
| Forgetting to consider all possible outcomes | List the full sample space |
ICSE Exam Focus
Probability carries 6–8 marks in ICSE exams. Question types:
- Single coin, die, or card problems.
- Two-coin or two-dice problems.
- Problems involving 'at least' or 'at most'.
- Real-life probability scenarios.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Single event probability (coin/die/card) | 3 |
| Two coins / two dice | 3 |
| Card problems | 3 |
| Real-life probability scenarios | 2 |
Self-Test Questions
-
A coin is tossed three times. Find the probability of getting (a) exactly two heads, (b) at least two tails.
-
A card is drawn from a well-shuffled pack of 52 cards. Find the probability that it is (a) a king, (b) a black card, (c) a red face card.
-
Two dice are rolled simultaneously. Find the probability that (a) both numbers are equal, (b) the product of numbers is 12.
-
A bag contains 5 red, 4 blue, and 3 green balls. One ball is drawn at random. Find the probability that it is (a) blue, (b) not red.
-
Find the probability that a leap year has 53 Sundays.
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Define (a) sample space, (b) event, (c) impossible event. Give an example of each.
In ICSE, probability answers must always be expressed as fractions in the simplest form. Avoid decimal approximations unless specifically asked.
