Lesson 19: What is Probability?

§Text Passage

1Summary 重點摘要

Probability measures how likely an event is. Rolling a die, getting 1 has probability 16. Learn to use fractions.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Coin FlipTossing a fair coin gives two results, heads and tails, each equally likely. The probability of heads = 12 = 50%, and the probability of tails is also 12. The two probabilities add up to 1, because every toss must show one of the two faces and there is no third possibility. Why a fair coin? Because the two sides weigh the same, so both are equally likely; with a lopsided coin the probabilities would be wrong, and experiments need a fair coin to mean anything. A common error is expecting one head in two tosses for sure; tosses are independent, the first result does not affect the second, and two tails in a row is entirely possible.
2. Block 2: Rolling a DieA six-sided die shows the numbers 1 to 6, each equally likely. The probability of rolling a 3 is 16, because only one of the six results is a 3. For a number greater than 4, the qualifying results are 5 and 6, two of them, so the probability is 26, which simplifies to 13. Why simplify? Because probability answers are conventionally written in simplest form: 26 and 13 are the same size, but the latter is the standard. A common error is miscounting the qualifying results, such as taking 'greater than 4' as three values by including 4; 'greater than 4' excludes 4, so list 5 and 6 one by one and count, and nothing extra slips in.
3. Block 3: Picking BallsA bag holds 4 red balls, 3 blue balls and 3 yellow balls, 10 in total. Drawing one at random, every ball has the same chance. The probability of red = 410 = 25; the probability of blue or yellow = (3 + 3)/10 = 610 = 35. Why count the total first? Because probability = favourable results ÷ total results, and the total is the denominator; a wrong denominator spoils every probability, which makes this the most error-prone step. A common error is missing one colour and taking the total as 7; list each colour's count, add them as 4 + 3 + 3 = 10, and only then compute the probabilities.
4. Block 4: Probability 0 and 1A probability of 0 means an impossible event, such as rolling a 7 on a six-sided die, since the die holds only 1 to 6 and no 7 exists among its results. A probability of 1 means a certain event, such as rolling one of the numbers 1 to 6, where all six results qualify, giving 66 = 1. The closer a probability is to 1, the more likely the event. Why is probability confined between 0 and 1? Because 0 corresponds to no favourable results at all and 1 to all results favourable, and anything outside the range is meaningless. A common error is producing a probability greater than 1; whenever that appears, a calculation has certainly gone wrong.
5. Block 5: Simplify FractionsProbability answers are usually written as fractions in simplest form. For example, 36 simplifies to 12 by dividing both parts by 3; 410 simplifies to 25 by dividing by 2; and 28 simplifies to 14. Simplifying changes nothing but the writing, making it tidier, so remember to finish in simplest form. Why simplify? Because the simplest form is the standard notation, easy to compare and communicate, and tests award marks by the convention. A common error is simplifying incompletely, stopping at 69 for 1218 when one more division by 3 gives 23; divide top and bottom by common factors until nothing more can be divided.
6. Block 6: More Trials = More AccurateTossing a coin 10 times in reality may not give exactly 5 heads; perhaps 6, perhaps 4. But the more times the coin is tossed, say 10000, the closer the share of heads comes to one half. This phenomenon is called the law of large numbers, studied in detail in secondary school; for now, remember that more trials bring steadier results. Why do more trials give more accuracy? Because chance deviations cancel out across many repetitions, the long-run proportion is held in place by the theoretical probability, and the influence of individual luck keeps shrinking. A common error is declaring the coin unfair after a few one-sided tosses; small samples deviate naturally, and only large-scale experiments support a conclusion.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Probability measures how likely an event is
• P(event) = favourable ÷ total
• Probability between 0 and 1
• 0 = impossible, 1 = certain
• Answers usually simplified