Chapter 14: Probability
This chapter is based on the classical (theoretical) definition of probability. Questions from it are scoring and appear every year — typically one 1-mark, one 2-mark and one 3-mark or case-study question built on coins, dice, playing cards, numbered tickets or defective items.
1. Theoretical (Classical) Probability
For an experiment whose outcomes are equally likely, the probability of an event E is:
P(E) = Number of outcomes favourable to E / Total number of possible outcomes
- 0 ≤ P(E) ≤ 1 for every event E. A probability can never be negative or greater than 1.
- Sure (certain) event: P(E) = 1, e.g. getting a number less than 7 on one throw of a die.
- Impossible event: P(E) = 0, e.g. getting a sum of 13 with two dice.
- Elementary event: an event having exactly one outcome. The sum of the probabilities of all elementary events of an experiment is 1.
2. Complementary Events
The event “not E” (written Ē) is the complement of E, and E and Ē are complementary events.
P(E) + P(not E) = 1, i.e. P(not E) = 1 − P(E)
Example: if P(winning) = 0.05, then P(not winning) = 1 − 0.05 = 0.95.
3. Standard Sample Spaces (memorise the totals)
| Experiment | Total outcomes | Sample space / structure |
|---|---|---|
| One coin | 2 | H, T |
| Two coins | 4 | HH, HT, TH, TT |
| Three coins | 8 | HHH, HHT, HTH, THH, HTT, THT, TTH, TTT |
| One die | 6 | 1, 2, 3, 4, 5, 6 |
| Two dice | 36 | ordered pairs (1,1), (1,2), …, (6,6) |
| One card from a deck | 52 | see structure below |
Structure of a deck of 52 playing cards
- 4 suits of 13 cards each: spades (♠, black), clubs (♣, black), hearts (♥, red), diamonds (♦, red).
- 26 red cards and 26 black cards.
- Each suit has 13 ranks: Ace, 2, 3, …, 10, Jack, Queen, King.
- Face cards: Jack, Queen, King — 3 per suit, so 12 face cards in all (6 red, 6 black). The Ace is not a face card.
- 4 cards of every rank: 4 aces, 4 kings, 4 queens, 4 jacks, etc.
4. Worked Examples (typical board patterns)
(a) Cards
One card is drawn from a well-shuffled deck of 52 cards. Then:
- P(a king) = 4/52 = 1/13
- P(a red face card) = 6/52 = 3/26
- P(neither a king nor a queen) = (52 − 8)/52 = 44/52 = 11/13
(b) Two dice (sums)
Two dice are thrown together; total outcomes = 36. Count favourable pairs by listing:
- P(sum = 7): (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) ⇒ 6/36 = 1/6
- P(sum = 8): (2,6), (3,5), (4,4), (5,3), (6,2) ⇒ 5/36
- P(doublet) = 6/36 = 1/6
(c) Defective items
A lot has 20 bulbs of which 4 are defective. One bulb is drawn at random.
P(defective) = 4/20 = 1/5. If the bulb drawn is not defective and is not replaced, then for the next draw the lot has 19 bulbs with 4 defective, so P(not defective now) = 15/19.
(d) Numbered balls / tickets
A box contains tickets numbered 1 to 40; one ticket is drawn.
P(multiple of 5) = 8/40 = 1/5; P(perfect square) = 6/40 = 3/20 (squares: 1, 4, 9, 16, 25, 36).
(e) At-least / at-most questions
Three coins are tossed together (8 outcomes).
- P(at least two heads) = P(2 or 3 heads) = {HHT, HTH, THH, HHH} ⇒ 4/8 = 1/2
- P(at most one head) = P(0 or 1 head) = {TTT, HTT, THT, TTH} ⇒ 4/8 = 1/2
Exam tip: “At least k” means k or more; “at most k” means k or fewer. When the counting is long, use the complement: P(at least one head) = 1 − P(no head). For two-dice questions, always write “Total outcomes = 36” and list the favourable ordered pairs — listing carries marks.
5. Stepwise Method for Any Probability Question
- Write the total number of equally likely outcomes.
- Define the event E clearly in words.
- List or count the outcomes favourable to E.
- Apply P(E) = favourable/total and simplify the fraction to lowest terms.
6. Quick Facts for 1-Mark Questions
- Probability of a sure event = 1; of an impossible event = 0.
- P(E) can be 0, 1 or any fraction/decimal between them — never negative, never more than 1.
- Prime numbers on a die: 2, 3, 5 ⇒ P(prime) = 3/6 = 1/2. (Note: 1 is neither prime nor composite.)
- If an ordinary year has 365 days = 52 weeks + 1 day, P(53 Sundays) = 1/7; a leap year has 52 weeks + 2 days, so P(53 Sundays) = 2/7.
Common mistakes to avoid: (1) Counting the Ace as a face card — it is not. (2) Treating (2,5) and (5,2) as one outcome with two dice — they are different ordered pairs. (3) Forgetting to reduce the total (52 → fewer cards) when some cards are stated to be removed from the deck. (4) Writing probability as a ratio like 3:26 — always write it as a fraction, decimal or per cent.
Rationalised syllabus note
No topics have been deleted from this chapter in the rationalised syllabus. The entire chapter (theoretical probability) is examinable.