Basic Probability

1Summary 重點摘要

Basic probability concepts, calculate simple event probability.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Know probabilityProbability is a number that describes how likely an event is. It always lies between 0 and 1: 0 means the event is impossible, and 1 means it is certain. The closer the value is to 1, the more likely the event is. For example, sunrise tomorrow is certain, while rolling a 7 on an ordinary die is impossible. Knowing this range helps you judge whether an answer is reasonable.
2. Step 2: Count total possibilitiesBefore calculating, list every outcome that can happen; this collection is called the sample space, and its size becomes the denominator of the formula. A fair die gives 6 outcomes, and drawing one ball from a bag of four balls gives 4. The outcomes must be equally likely, otherwise the simple formula cannot be used directly.
3. Step 3: Count favourable outcomesFavourable outcomes are the outcomes that match the event, and their number becomes the numerator. If the event is rolling an even number, then 2, 4 and 6 all match, so there are 3 favourable outcomes; if the event is a number greater than 4, only 5 and 6 match. List them one by one so that nothing is counted twice or missed.
4. Step 4: Probability = favourable / totalWrite the number of favourable outcomes over the total number of outcomes: P(event) = favourable outcomes ÷ total outcomes. For an even number on a fair die, P(even) = 36. Remember that this formula only works when every outcome is equally likely, such as with a fair die or a well-shuffled pack of cards.
5. Step 5: Write a fraction or a decimalA probability is usually written as a fraction in its simplest form, so 36 is simplified to 12. It can also be written as a decimal between 0 and 1, such as 0.5, or as a percentage, such as 50%. Unless a question states otherwise, the simplest fraction is the safest form in an exam, because it clearly shows the numerator and the denominator.
6. Step 6: Check your answerAfter calculating, check that the answer lies between 0 and 1; a probability greater than 1 or a negative one means a mistake has been made. Then use complementary events: the probability of an event plus the probability of its complement should equal exactly 1. For example, P(even) = 12 and P(odd) = 12, and 12 + 12 = 1, so the calculation is sound.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Probability from 0 to 1
• P = favorable / total outcomes
• 0 = impossible, 1 = certain
• List all possible outcomes first
• Probability cannot exceed 1