Probability Basics

1Summary 重點摘要

Probability P(event) = number of favourable outcomes ÷ total number of equally likely outcomes. It ranges from 0 (impossible) to 1 (certain). All mutually exclusive outcomes must add up to 1.

2Vocabulary cards Vocabulary

3Step by step Steps

1. What probability is, and its rangeProbability measures how likely an event is, always between 0 and 1: 0 impossible, 1 certain, 0.5 an even chance, also writable as a percentage (0.5 = 50%). Everyday anchors: drawing a white ball from a bag holding only red ones has probability 0; a fair coin gives heads with probability 0.5; the sun rising in the east has probability 1. Any result above 1 or below 0 is certainly wrong — re-check at once whether numerator and denominator were swapped or the favourable outcomes miscounted.
2. The basic formulaWhen every outcome is equally likely, P(event) = favourable outcomes ÷ all possible outcomes. Rolling a fair die, the favourable outcome is {6} — one of them; all outcomes are {1, 2, 3, 4, 5, 6} — six in all; hence P(6) = 1 ÷ 6. The formula presumes every outcome equally likely: the die must be fair. A loaded die gives some faces a better chance than others, the assumption breaks, and the formula no longer applies.
3. Example: rolling an even numberFor a fair die, the favourable outcomes for an even number are {2, 4, 6} — three of them — against six outcomes in all, so P(even) = 3 ÷ 6 = 12 = 0.5 = 50%. The same probability may be written as the fraction 12, the decimal 0.5 or 50%, all equivalent; follow the form the question requests, or give all three. List favourable outcomes one by one to avoid mental slips: 'greater than 4' means {5, 6}, two outcomes, not one.
4. Complement: subtract from 1Sometimes the probability of 'not happening' is far easier than that of 'happening'; then use the complement rule P(not A) = 1 − P(A). Since P(6) = 16, the probability of not rolling a six is 1 − 16 = 56. An event and its complement together cover every possibility, so their probabilities sum to exactly 1. Whenever a question says 'at least once' or 'not', ask first whether the complement gives a shorter path — for at-least-one-six problems it usually does.
5. The playing-card exampleFrom a standard 52-card deck (no jokers), one card is drawn at random: 13 hearts give P(heart) = 13 ÷ 52 = 14, and four aces give P(A) = 4 ÷ 52 = 113. The key is knowing the deck's structure: 13 cards per suit and four cards of each rank must be memorised. If a condition is attached — say 'given the card is a heart, find the probability it is the ace of hearts' — the denominator shrinks to 13, a first taste of conditional probability met formally in senior forms.
6. Independence and the gambler's fallacyFour essentials: probability lies between 0 and 1; all outcomes' probabilities sum to 1; fair means every outcome equally likely; and successive trials are independent. The commonest misconception is the gambler's fallacy — after five sixes in a row, believing the next roll cannot be a six. A die has no memory: the chance is still 16. Likewise a lottery number drawn twice in a row is no less likely next time. Understanding independence guards against many traps in everyday judgement.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• P = favourable ÷ all outcomes
• Probability is 0 to 1
• Not happening = 1 − happening
• Each trial independent, don't be affected by previous results
• Fair = every outcome equally likely