Lesson 20: Probability Applications

§Text Passage

1Summary 重點摘要

Expected = probability × total. Roll die 60 times, expected 1s? = 16 × 60 = 10. Learn to predict using probability.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Expected Frequency FormulaExpected frequency = probability × total number of trials; it predicts roughly how often a result appears over many trials. For example, rolling a die 30 times, how many sixes are expected? The probability of a six is 16, so the expected frequency = 16 × 30 = 5 times. Why may it be predicted this way? Because each roll lands on 6 with a one-in-six chance, and over 30 rolls about one sixth of them should fall on 6 in theory. It is a theoretical average, not a promise. A common error is treating the expected frequency as a guaranteed count; the expectation only describes the long-run average, and results may fall above or below it, so answers carry the word expected.
2. Block 2: Picking Balls PredictionFor example, a bag holds 3 red and 7 blue balls, 10 in total. Each ball drawn is put back, and this repeats 50 times; how many red draws are expected? The probability of red is 310, so the expectation = 310 × 50 = 15 times. Why watch the 'with replacement' condition? Because replacing keeps the bag's make-up unchanged, so every draw faces the same probability of 310 and the formula holds; without replacement the red balls would dwindle and the probability would shift, breaking the calculation. A common error is applying the formula without checking the condition; confirm that every trial runs under the same conditions before using the expectation.
3. Block 3: Probability as PercentageWeather reports often state probability as a percentage: a 70% chance of rain means that in similar weather situations, about 7 days in 10 bring rain. For example, over the coming 30 days how many rainy days are expected? Expected days = 70% × 30 = 21 days. Multiplying the percentage by the total gives the expected count or number of days. Why may the percentage multiply directly? Because 70% is 0.7, and 0.7 × 30 = 21, exactly matching the fraction method. A common error is skipping the conversion and computing 70 × 30 = 2100; convert first, then multiply, and afterwards check that the day count does not exceed the total days.
4. Block 4: Not ExactThe expected frequency is a theoretical average, and the real result need not match it exactly. For example, tossing a coin 10 times may give 4 heads instead of exactly 5; this is normal fluctuation, not a computational error. Why do more trials make predictions more reliable? Because the more times the experiment repeats, the closer the actual proportion comes to the theoretical probability, with chance deviations thinned out across the whole; over 1000 tosses, the share of heads sits very near one half. A common error is judging the probability wrong whenever a result departs from the expectation; short-run deviations are normal, and the working, not one result, decides correctness.
5. Block 5: Addition Rule (Mutually Exclusive)Two events that cannot happen at the same time are mutually exclusive, and the probability that one of them happens is the sum of the two probabilities. For example, rolling a die, the probability of a 1 or a 2 = 16 + 16 = 26 = 13, because one roll cannot show both 1 and 2; the two events are exclusive. Why may the probabilities add? Because the two results each hold their own separate share of the chances, and together they hold two shares, so the chances merge. A common error is adding probabilities for events that can co-occur, such as 'an even number or a 2', where 2 gets counted twice; first test whether the events can happen together, and if they can, do not simply add.
6. Block 6: Word Problem: Lucky DrawWord problem: in a lucky draw the chance of winning is 15; playing 40 times, how many wins are expected? Expected frequency = 15 × 40 = 8 times. Write '8 wins are expected', understanding that the real number may be more or fewer than 8, since the expectation is only a long-run average. Why does this kind of question matter in life? Because shops' promotions and game designs all predict results with probability, and reading the expected value lets one judge rationally; a 15 chance does not mean one win is guaranteed in five plays. A common error is treating the expectation as a promise, or forgetting to multiply by the total; keep the word expected in the answer and state the unit of times.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Expected = probability × total
• Probability can predict
• Actual may differ from expected
• More trials → closer to expected
• Mutually exclusive: add probabilities