AGES 8–10 · CHANCE AND POSSIBLE OUTCOMES
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Separate what could happen from what happened, and explain the equal chances behind a claim.
Chapter index · Worked correction · Discussion and checking · Vocabulary articles
A surprising result is a useful place to learn a precise word. If a bag has three red tokens and one blue token, drawing blue is surprising but not a mistake. This guide helps children aged roughly 8–10 describe what could happen, what actually happened and what the evidence supports. The age band is a guide, not a readiness test; an adult can read the story aloud and begin with only five words.
All four editions share the same numbered meanings, examples and original story. Use the links below a word to move to its matching card in another language. Learn one meaning well before collecting four labels. These are context-matched expressions, not a claim that every word has one translation in every sentence.
Mandarin uses Simplified Chinese and tone-marked pinyin. Hindi and Russian include approximate Roman reading aids; Russian stress marks are learning supports and are usually omitted in ordinary writing. Listen to a competent speaker where possible. The lesson uses imagined token draws, not betting, prizes, personal predictions or a claim about a national curriculum.
Chapter index
1–4 · Count the equal chances, not just the labels
5–8 · Procedures and predictions: words 11–15
9–12 · A gentle seven-day practice plan
CHAPTER 1 OF 12 · Understand the ideas
1. Count the equal chances, not just the labels
Begin with the setup, not with a guess. Imagine four tokens identical in size, shape, mass and feel, differing only in colour. Bag A contains three red and one blue. An ideal draw gives each individual token the same chance. A colour groups several token outcomes together: red has three ways to occur and blue only one. Two colour names do not mean two equal chances.
Worked correction
Worked correction: “There are two colours, so red and blue must each have half the chance.” Keep the correct observation: there are two possible colour labels. Repair the reasoning by counting the four equally likely individual tokens. Red has 3 out of 4 chances; blue has 1 out of 4. The fractions describe the model, not a promise that four draws must contain exactly three reds and one blue.
For every trial, draw without choosing a colour, record it, return the token and mix again. We assume independent ideal draws: the previous result does not alter the next chance. Without replacement, the contents change and the next chance may change too. Keep those two procedures separate. Use large safe classroom tokens if acting this out; younger children should not handle small loose pieces.
Read the map by meaning rather than by sound alone. “Certain” is stronger than “likely”, while “unlikely” still permits the event. Pair a positive sentence with a boundary: blue is possible in A because it is present; green is impossible because the stipulated bag has none. If someone changes the contents, revisit the statements instead of treating them as universal truths.
Some entries are phrases because that is the natural teaching expression. In English “chance” can also mean an opportunity; here it means likelihood. Russian достоверный here describes a certain event, not a trustworthy person. Hindi बारंबारता and Russian частота here refer to an absolute count in a stated series. Always include the series size when discussing the count.
1. chance · 可能性
English: chance
How likely something is to happen; not a promise about the next try.
Red has a greater chance than blue in bag A.
中文（普通话）: 可能性 (kěnéngxìng)
某件事发生的可能程度，不是对下一次的保证。
A袋抽到红色的可能性比蓝色大。
हिन्दी: संभावना (sambhāvnā)
किसी बात के होने की संभावना; अगली बार का वादा नहीं।
थैली A में लाल की संभावना नीले से अधिक है।
Русский: вероя́тность (veroyatnost)
Насколько вероятно событие; не обещание результата следующей попытки.
В мешке A красный вероятнее синего.
Try it: Say “chance” without promising the next result.
2. outcome · 结果
English: outcome
One result that can occur in a trial, described at the level we choose to record.
Our recorded outcome is the colour, not the token’s number.
中文（普通话）: 结果 (jiéguǒ)
一次试验可能出现的一种结果，需说清记录什么。
我们记录的结果是颜色，不是片号。
हिन्दी: परिणाम (pariṇām)
एक परीक्षण में आने वाला परिणाम; बताएँ कि क्या दर्ज कर रहे हैं।
हम रंग दर्ज करते हैं, गोटी की संख्या नहीं।
Русский: исхо́д (iskhod)
Один возможный результат попытки на выбранном уровне описания.
Мы записываем цвет, а не номер жетона.
Try it: Name what is recorded: colour or individual token?
3. possible · 可能的
English: possible
Able to happen under the stated rules, even if it has not happened yet.
Blue is possible because one blue token is in A.
中文（普通话）: 可能的 (kěnéng de)
在给定规则下能发生，即使尚未发生。
A有一枚蓝片，所以蓝色可能出现。
हिन्दी: संभव (sambhav)
तय नियमों में हो सकता है, भले अभी न हुआ हो।
A में एक नीली गोटी है, इसलिए नीला संभव है।
Русский: возмо́жный (vozmozhnyy)
Может произойти по заданным правилам, даже если ещё не происходило.
Синий возможен: в A есть один синий жетон.
Try it: Explain why blue remains possible after three reds.
4. impossible · 不可能的
English: impossible
Unable to happen within the stated setup; check the conditions before using this strong word.
Green is impossible when A contains only red and blue tokens.
中文（普通话）: 不可能的 (bù kěnéng de)
在给定条件下不能发生，使用这个词前先核对条件。
A只有红蓝片时，绿色不可能出现。
हिन्दी: असंभव (asambhav)
तय व्यवस्था में नहीं हो सकता; यह शब्द कहने से पहले शर्तें जाँचें।
A में केवल लाल-नीली गोटियाँ हों तो हरा असंभव है।
Русский: невозмо́жный (nevozmozhnyy)
Не может произойти в указанной модели; сначала проверьте условия.
Зелёный невозможен, если в A только красные и синие жетоны.
Try it: Give the condition that makes green impossible.
5. certain · 必然的
English: certain
Guaranteed within the given model; every allowed outcome makes the statement true.
A draw from A is certain to be red or blue.
中文（普通话）: 必然的 (bìrán de)
在给定模型中一定发生，所有允许结果都符合。
从A抽一片，必然是红或蓝。
हिन्दी: निश्चित (niścit)
दिए मॉडल में पक्का; हर अनुमत परिणाम में बात सही हो।
A से निकली गोटी का लाल या नीला होना निश्चित है।
Русский: достове́рный (dostovernyy)
Обязательно в данной модели: утверждение верно при любом допустимом исходе.
Жетон из A достоверно будет красным или синим.
Try it: Complete: red or blue is certain because ___.
6. uncertain · 不确定的
English: uncertain
Not known for sure; uncertainty does not mean that every possibility has the same chance.
The next colour is uncertain, although red is more likely.
中文（普通话）: 不确定的 (bù quèdìng de)
不能肯定；不确定不代表各种结果机会相等。
下次颜色不确定，虽然红色更可能。
हिन्दी: अनिश्चित (aniścit)
पक्का ज्ञात नहीं; इसका अर्थ सभी संभावनाएँ बराबर होना नहीं।
अगला रंग अनिश्चित है, हालाँकि लाल अधिक संभावित है।
Русский: неопределённый (neopredelyonnyy)
Заранее не известен точно; это не означает равенства всех шансов.
Следующий цвет неопределён, хотя красный вероятнее.
Try it: Distinguish “unknown next colour” from “equal colour chances”.
7. likely · 很可能的
English: likely
Expected more readily; in this lesson’s two-colour examples it means a chance greater than one half.
Red is likely in A, but blue can still occur.
中文（普通话）: 很可能的 (hěn kěnéng de)
较有可能；本课两色情境指机会超过一半。
A中红色很可能出现，但仍可出现蓝色。
हिन्दी: अधिक संभावित (adhik sambhāvit)
जिसकी उम्मीद अधिक हो; इस दो-रंग पाठ में आधे से अधिक संभावना।
A में लाल अधिक संभावित है, पर नीला आ सकता है।
Русский: вероя́тный (veroyatnyy)
Ожидаемый с большей вероятностью; здесь для двух цветов — шанс больше половины.
В A красный вероятен, но синий тоже может выпасть.
Try it: Give a likely event that can still fail.
8. unlikely · 不太可能的
English: unlikely
Having a small chance, not no chance; here blue has one chance out of four.
Blue is unlikely in A, not impossible.
中文（普通话）: 不太可能的 (bù tài kěnéng de)
机会较小，不是没有机会；这里蓝色占四份之一。
A中蓝色不太可能，不是不可能。
हिन्दी: कम संभावित (kam sambhāvit)
कम संभावना, शून्य नहीं; यहाँ नीले का चार में एक मौका है।
A में नीला कम संभावित है, असंभव नहीं।
Русский: маловероя́тный (maloveroyatnyy)
С небольшим, но не нулевым шансом; здесь синий имеет один шанс из четырёх.
Синий в A маловероятен, но не невозможен.
Try it: Replace “blue cannot happen” with an accurate phrase.
9. equally likely · 可能性相等
English: equally likely
Having the same chance under the same rules; equal numbers of labels alone do not establish this.
Red and blue are equally likely in B with two tokens of each.
中文（普通话）: 可能性相等 (kěnéngxìng xiāngděng)
在相同规则下机会一样；类别名称数量相同并不能证明。
B各有两片红蓝，红蓝可能性相等。
हिन्दी: समान संभावित (samān sambhāvit)
समान नियमों में बराबर संभावना; केवल नाम गिनना पर्याप्त नहीं।
B में दो-दो लाल-नीली गोटियाँ हैं, इसलिए दोनों समान संभावित हैं।
Русский: равновероя́тный (ravnoveroyatnyy)
Имеющий такой же шанс при тех же правилах; числа названий недостаточно.
В B по два жетона каждого цвета, поэтому цвета равновероятны.
Try it: Compare each single token’s chance with the red group’s chance.
10. fair · 公平的
English: fair
Meeting the agreed fairness rule; our colour game calls equal chances of winning fair.
B is fair for a one-draw red-versus-blue game.
中文（普通话）: 公平的 (gōngpíng de)
符合约定的公平规则；本课颜色游戏以双方获胜机会相等为公平。
单次红对蓝的游戏用B公平。
हिन्दी: निष्पक्ष (niṣpakṣ)
तय निष्पक्षता-नियम पूरा करने वाला; इस खेल में जीत के बराबर मौके।
एक बार लाल-बनाम-नीला खेल में B निष्पक्ष है।
Русский: справедли́вый (spravedlivyy)
Соответствующий оговорённому правилу: здесь равные шансы игроков на победу.
B справедлив для игры «красный против синего» с одним вытягиванием.
Try it: State the game’s fairness rule before naming a fair bag.
11. unfair · 不公平的
English: unfair
Not meeting the agreed fairness rule; unequal chances can come from a setup, not a person’s bad intention.
A is unfair for that game because red has three tokens and blue one.
中文（普通话）: 不公平的 (bù gōngpíng de)
不符合约定公平规则；机会不等可来自设置，不一定是有人故意。
该游戏用A不公平，因为红三片、蓝一片。
हिन्दी: पक्षपाती (pakṣpātī)
तय निष्पक्षता-नियम से हटने वाला; असमान मौके व्यवस्था से भी आ सकते हैं।
उस खेल में A पक्षपाती है: तीन लाल, एक नीली।
Русский: несправедли́вый (nespravedlivyy)
Не соответствующий правилу равных шансов; причина может быть в устройстве, а не в намерении человека.
Для этой игры A несправедлив: три красных жетона и один синий.
Try it: Describe an unfair setup without accusing a player.
12. random · 随机的
English: random
Determined through a chance process rather than deliberately choosing the result; random need not mean equal chances for colour groups.
A random token draw can favour red as a colour.
中文（普通话）: 随机的 (suíjī de)
由机会过程决定，不是刻意挑结果；随机不等于各颜色组机会相同。
随机抽一片，仍可更容易抽到红色。
हिन्दी: यादृच्छिक (yādṛcchik)
जानबूझकर परिणाम चुनने के बजाय संयोग की प्रक्रिया से तय; रंग-समूह बराबर होना ज़रूरी नहीं।
यादृच्छिक गोटी निकालने पर भी लाल रंग अधिक संभावित हो सकता है।
Русский: случа́йный (sluchaynyy)
Определяемый случайным процессом, а не выбором нужного результата; шансы групп цветов могут различаться.
При случайном выборе жетона красный цвет может быть вероятнее.
Try it: Explain how a random draw can favour a colour.
13. predict · 预测
English: predict
Say what you expect before observing, using a reason that can later be checked.
I predict red more often over many trials, not red on every trial.
中文（普通话）: 预测 (yùcè)
观察前根据理由说预期，之后可以检查。
我预测很多次中红色较多，不是每次红色。
हिन्दी: पूर्वानुमान लगाना (pūrvānumān lagānā)
देखने से पहले कारण सहित उम्मीद बताना, जिसे बाद में जाँचा जा सके।
कई परीक्षणों में लाल अधिक आने का पूर्वानुमान है, हर बार लाल नहीं।
Русский: предска́зывать (predskazyvat)
До наблюдения сказать, чего ожидаете, и назвать проверяемую причину.
Я предсказываю больше красных за много попыток, а не красный каждый раз.
Try it: Make a reasoned prediction and add “not guaranteed”.
14. trial · 试验
English: trial
One complete go using the agreed procedure; here draw one token, record, replace and mix.
Our first trial produced blue.
中文（普通话）: 试验 (shìyàn)
按约定流程完成一次；本课抽一片、记录、放回、混匀。
第一次试验抽到蓝色。
हिन्दी: परीक्षण (parīkṣaṇ)
तय प्रक्रिया का एक पूरा दौर: निकालें, दर्ज करें, लौटाएँ, मिलाएँ।
पहले परीक्षण में नीला आया।
Русский: испыта́ние (ispytaniye)
Одна полная попытка по процедуре: вытянуть, записать, вернуть и перемешать.
Первое испытание дало синий цвет.
Try it: Say all four trial steps in order.
15. repeat · 重复
English: repeat
Carry out the procedure again; state what is kept the same.
Replace the token before you repeat the trial.
中文（普通话）: 重复 (chóngfù)
再次执行流程，并说清哪些条件不变。
重复试验前先放回片。
हिन्दी: दोहराना (dohrānā)
प्रक्रिया फिर करना; बताएँ क्या समान रखा जाता है।
परीक्षण दोहराने से पहले गोटी लौटाएँ।
Русский: повторя́ть (povtoryat)
Выполнить процедуру снова, указав сохраняемые условия.
Верните жетон, прежде чем повторять испытание.
Try it: Explain why replacement matters before repeating.
16. record · 记录
English: record
Keep an accurate account, including surprising results instead of changing them to fit a prediction.
Record blue even when you expected red.
中文（普通话）: 记录 (jìlù)
准确保留结果，包括意外结果，不为迎合预测改动。
即使预期红色，也如实记录蓝色。
हिन्दी: दर्ज करना (darj karnā)
सही विवरण रखना; चौंकाने वाले परिणाम भी लिखना, अनुमान से मिलाने को बदलना नहीं।
लाल की उम्मीद थी, फिर भी नीला दर्ज करें।
Русский: запи́сывать (zapisyvat)
Точно сохранять наблюдения, включая неожиданные, не подгоняя их под прогноз.
Запишите синий, даже если ожидали красный.
Try it: Record the supplied sequence without improving its pattern.
17. frequency · 频数
English: frequency
The number of times an outcome occurs in a stated set of trials; our tables use counts, not percentages.
In our four-trial record the frequency of blue is two.
中文（普通话）: 频数 (pínshù)
在指定一组试验中出现的次数；本课表格用次数，不是百分比。
四次记录中蓝色频数是二。
हिन्दी: बारंबारता (bārambārtā)
तय परीक्षण-समूह में परिणाम आने की गिनती; यहाँ प्रतिशत नहीं।
चार परीक्षणों में नीले की बारंबारता दो है।
Русский: частота́ (chastota)
Число появлений исхода в заданной серии; здесь это количество, не процент.
В записи четырёх испытаний частота синего равна двум.
Try it: Count blues and name the total trial count too.
18. compare · 比较
English: compare
Examine similarities and differences while keeping the question and procedure clear.
Compare bag contents separately from the short result record.
中文（普通话）: 比较 (bǐjiào)
问题和流程清楚时，检查相同与不同。
分别比较袋内组成与短期记录。
हिन्दी: तुलना करना (tulnā karnā)
प्रश्न और प्रक्रिया स्पष्ट रखकर समानताएँ-अंतर देखना।
थैली की सामग्री और छोटे परिणाम-अभिलेख की अलग तुलना करें।
Русский: сра́внивать (sravnivat)
Находить сходства и различия при ясно заданных вопросе и процедуре.
Сравнивайте состав мешков отдельно от короткой записи результатов.
Try it: Compare A and B using contents, not just recent results.
19. evidence · 证据
English: evidence
Information that supports reasoning; explain what it can and cannot establish.
Two blues are evidence of what happened, not proof that blue is due next.
中文（普通话）: 证据 (zhèngjù)
支持推理的信息，也要解释它能说明与不能说明什么。
两次蓝是已发生的证据，不证明下次该轮到蓝。
हिन्दी: साक्ष्य (sākṣya)
तर्क को सहारा देने वाली जानकारी; बताएँ इससे क्या स्थापित होता है और क्या नहीं।
दो नीले हुए का साक्ष्य हैं, अगला नीला होना तय नहीं।
Русский: свиде́тельство (svidetelstvo)
Информация для обоснования; важно указать, что она подтверждает, а чего не доказывает.
Два синих — свидетельство произошедшего, не доказательство, что следующий обязан быть синим.
Try it: Say one thing the four-trial record cannot prove.
20. conclude · 得出结论
English: conclude
State what follows from the information, with its limits and conditions.
We conclude blue is possible, not that both bags have equal colour chances.
中文（普通话）: 得出结论 (déchū jiélùn)
说出信息支持什么，并带上范围与条件。
我们得出蓝色可能出现，不是两袋颜色机会相同。
हिन्दी: निष्कर्ष निकालना (niṣkarṣ nikālnā)
जानकारी से मिलने वाला निष्कर्ष, उसकी सीमा और शर्तों सहित बताना।
निष्कर्ष है कि नीला संभव है, दोनों थैलियों के रंग-मौके बराबर नहीं।
Русский: де́лать вы́вод (delat vyvod)
Сформулировать, что следует из информации, сохраняя условия и ограничения.
Мы делаем вывод: синий возможен, но шансы цветов в двух мешках не одинаковы.
Try it: Conclude with a condition: “Under this model…”.
The following story and every numerical record are original fictional teaching examples. No real class results are claimed. The ideal token model is an assumption, not proof that any household bag mixes perfectly. The initial composition table and the observed-trial table answer different questions: what chances the model assigns, and what this short sequence happened to show.
Follow two kinds of number without mixing them. Bag A has four physical tokens: three red and one blue. The record also has four trials, but those are four selections, not four tokens removed permanently. Because the token is returned each time, the same blue token can be selected twice or more. Equal observed counts in a short run do not turn a 3-to-1 composition into a 2-to-2 composition.
| Bag | Red tokens | Blue tokens | Red chance | Blue chance |
|---|---|---|---|---|
| A | 3 | 1 | 3/4 | 1/4 |
| B | 2 | 2 | 2/4 = 1/2 | 2/4 = 1/2 |
| Trial | Observed colour | Bag after replacement |
|---|---|---|
| 1 | Blue | 3 red, 1 blue |
| 2 | Red | 3 red, 1 blue |
| 3 | Blue | 3 red, 1 blue |
| 4 | Red | 3 red, 1 blue |
English · The Two Colour Names
Nila drew two circles on a page and labelled them RED and BLUE. “Two colours means half each,” she said. Omar looked at bag A’s list: three red tokens and one blue, all otherwise identical. “Let’s count the tokens that can give each colour,” he suggested.
They imagined an ideal draw in which each of the four tokens had the same chance. Three tokens led to red; one led to blue. Nila kept her two colour labels but changed her explanation. “Red is more likely. Blue is possible, though unlikely. Green is impossible with these contents.”
For their story record they used the sequence blue, red, blue, red. After each draw they returned the token and mixed again. Blue appeared twice in four trials. “The count is two blues and two reds,” said Omar. “But the bag still has three red tokens and one blue.”
Nila almost said the bag had become fair. Then she checked the agreed game rule: one player wins on red, the other on blue, with one draw deciding a round. Under the model, bag A favoured the red player. A short tied record did not give the players equal chances.
They checked bag B, which had two red and two blue tokens. With the same ideal procedure, B gave equal colour chances. Even so, four draws from B did not have to produce two of each. Fairness of chances was not a promise of an evenly split short record.
“Suppose red appeared three times in a row,” Nila asked. “Would blue be due?” Omar pointed to the replacement step. “The bag resets each time. In our independent model, blue would still have one chance out of four in A. It can happen next, but it does not have to.”
Their final sentence was careful: “We can predict what is more likely, record what happened and change a claim when its reasoning fails. We do not change a result just because it surprises us.”
中文（普通话） · 两个颜色名称
妮拉在纸上画两个圆，写红和蓝。“两种颜色，就是各一半。”奥马尔看A袋清单：三红一蓝，其他特征相同。“数数哪些片能给出每种颜色吧。”
他们想象理想抽取，四片各有同样机会。三片通向红色，一片通向蓝色。妮拉保留两个颜色名称，却改了解释：“红色更可能。蓝色可能，但不太可能。这个组成没有绿，所以绿色不可能。”
故事记录采用蓝、红、蓝、红。每次抽后放回并混匀。四次中蓝出现两次。奥马尔说：“记录二蓝二红，袋内仍三红一蓝。”四次试验不是取走四片，同一蓝片能再抽到。
妮拉差点说袋变公平了，便重看约定游戏：一人选红，一人选蓝，一抽定一轮。模型中A偏向红方。短期打平不让双方机会变相等。
再看B袋，两红两蓝，同样理想流程下颜色机会相等。不过B抽四次也不必二红二蓝。机会公平，不保证短期记录平均分配。
“若连出三红，下次该轮蓝吗？”妮拉问。奥马尔指放回步骤：“每次恢复同样组成。在独立模型中，A的蓝仍占四份之一。下次能出蓝，却不是必须蓝。”
他们最后写：“可预测较可能什么，如实记实际结果，推理有错就改陈述。不因结果意外而改结果。”他们用另一语言的同号词卡再说一遍，先保留意思，再换词。
हिन्दी · दो रंगों के नाम
नीला ने कागज़ पर दो गोले बनाकर लाल और नीला लिखा। ‘दो रंग मतलब आधा-आधा,’ उसने कहा। ओमर ने A की सूची देखी: तीन लाल गोटियाँ, एक नीली, बाकी सब समान। उसने कहा, ‘हर रंग देने वाली अलग गोटियाँ गिनें।’
उन्होंने आदर्श चयन सोचा जिसमें चारों गोटियों का समान मौका था। तीन से लाल, एक से नीला। नीला ने दोनों रंग-नाम रखे, पर व्याख्या सुधारी: ‘लाल अधिक संभावित है। नीला संभव है, मगर कम संभावित। इस सामग्री में हरा असंभव है।’
कहानी का क्रम नीला, लाल, नीला, लाल था। हर बार गोटी लौटाकर फिर मिलाया। चार परीक्षणों में नीला दो बार आया। ओमर बोला, ‘गिनती दो नीली और दो लाल है। पर थैली में अभी भी तीन लाल-एक नीली हैं।’ चार परीक्षण चार गोटियाँ हमेशा के लिए निकालना नहीं थे।
नीला लगभग कहने लगी कि थैली निष्पक्ष बन गई। फिर खेल का तय नियम देखा: एक खिलाड़ी लाल पर जीतेगा, दूसरा नीले पर; एक चयन से एक दौर तय होगा। मॉडल में A लाल खिलाड़ी के पक्ष में था। छोटा बराबरी वाला अभिलेख जीत के मौके बराबर नहीं करता।
B में दो लाल-दो नीली थीं। उसी आदर्श प्रक्रिया में रंगों के मौके बराबर थे। फिर भी B से चार बार निकालने पर ठीक दो-दो आना आवश्यक नहीं था। मौके निष्पक्ष होना छोटी श्रृंखला बराबर बँटने का वादा नहीं है।
‘अगर लगातार तीन लाल आएँ, तो अब नीला आना चाहिए?’ नीला ने पूछा। ओमर ने लौटाने वाला कदम दिखाया। ‘हर बार वही सामग्री फिर रहती है। स्वतंत्र मॉडल में A से नीले का मौका अब भी चार में एक होगा। अगला नीला हो सकता है, होना आवश्यक नहीं।’
आखिर उन्होंने लिखा: ‘हम अधिक संभावित का पूर्वानुमान लगाते हैं, जो हुआ उसे दर्ज करते हैं और तर्क गलत हो तो कथन सुधारते हैं। चौंकाने वाला परिणाम मिटाते नहीं।’
Русский · Два названия цвета
Нила нарисовала два кружка и подписала: КРАСНЫЙ и СИНИЙ. «Два цвета — значит, каждому половина», — сказала она. Омар посмотрел на состав A: три красных жетона и один синий, во всём остальном одинаковые. «Посчитаем, какие жетоны дают каждый цвет».
Они представили идеальный выбор, при котором каждый из четырёх жетонов равновероятен. Три давали красный, один — синий. Нила сохранила два названия, но исправила объяснение: «Красный вероятнее. Синий возможен, хотя маловероятен. Зелёный при этом составе невозможен».
Для вымышленной записи взяли последовательность: синий, красный, синий, красный. После каждой попытки жетон возвращали и снова перемешивали. Синий появился дважды за четыре испытания. «В записи два синих и два красных, — сказал Омар. — Но в мешке по-прежнему три красных и один синий». Четыре испытания не означали четыре навсегда вынутых жетона.
Нила чуть не сказала, что мешок стал справедливым. Затем проверила правило игры: один игрок побеждает на красном, второй на синем, раунд решается одним выбором. В модели A давал преимущество красному игроку. Короткая ничья в записи не уравнивала шансы.
В B было по два жетона каждого цвета. При той же идеальной процедуре цвета были равновероятны. Но и четыре выбора из B не обязаны дать по два каждого. Справедливые шансы не обещают ровного деления короткой серии.
«А если красный выпал три раза подряд, теперь очередь синего?» — спросила Нила. Омар показал шаг возвращения: «Каждый раз состав восстанавливается. В нашей независимой модели шанс синего в A остаётся один из четырёх. Он может выпасть следующим, но не обязан».
Их вывод был точным: «Мы предсказываем более вероятное, записываем случившееся и исправляем утверждение, если ошибочно рассуждение. Не переписываем результат лишь потому, что он удивил».
Answer aloud, point to a table or draw a labelled bag. A full explanation is more useful than a fast label. Ask the learner to complete “I think ___ because ___; this does not mean ___.” Keep the model assumptions visible when checking, and allow a home-language explanation before trying a new language.
1. Why are two colour labels not enough to claim equal chances?
2. Is blue impossible in A? Give evidence from the setup.
3. What is certain, and what is impossible, in one draw from A?
4. The record is blue, red, blue, red. What are the frequencies?
5. Does that tied record make A fair for the agreed colour game?
6. Must four draws from B contain exactly two reds?
7. After three reds from A, is blue due next?
8. What changes if A’s only blue token is removed and not replaced?
9. Which bag gives red a greater chance: C with 4 red and 2 blue, or D with 1 of each?
10. What would you check before trusting a physical token experiment?
Discussion guide
A’s four individual tokens are equally likely; three lead to red and one to blue. Count the equal elementary chances, not merely the groups.
No. One blue token is present. Its model chance is 1 out of 4, so it is possible but less likely than red.
Red or blue is certain. Green is impossible under the stated contents; these claims depend on the unchanged setup.
Blue 2 and red 2, across 4 trials. This describes observations, not a new bag composition.
No. The game asks for equal winning chances. A still gives red 3 out of 4 and blue 1 out of 4 under the ideal model.
No. Equal chances on each independent trial do not prescribe the counts of a short run.
No. With replacement and independent ideal draws, blue remains 1 out of 4. The record does not compensate itself.
Only three reds remain, so a next blue is impossible and red certain in that changed setup. This is not our replacement procedure.
C: 4/6 is greater than 1/2; pair the six C tokens into three pairs, two pairs red and one blue. D has equal colour chances.
Check the contents, indistinguishable feel, no looking or choosing, recording, replacement and mixing. A real procedure may not match the ideal model perfectly.
Five to ten calm minutes can be enough for one session; stop earlier if the child has had enough. This plan is flexible, not a promise of mastery in a week. Revisit a small contrast rather than drilling all twenty words daily. Use drawings if no safe tokens are available. There is no need to run hundreds of trials or to obtain a particular pattern.
| Day | Words | Small activity | What to notice |
|---|---|---|---|
| 1 | 1–5 | Draw A; name possible, impossible and certain statements. | Tie every strong statement to the contents. |
| 2 | 6–10 | Compare likely with certain; identify B’s equal chances. | Equal chances do not prescribe the next colour. |
| 3 | 11–15 | Explain the game rule and say the trial steps. | Random token choice can give unequal colour chances. |
| 4 | 16–20 | Read the four-trial record and state a limited conclusion. | Describe what happened without changing the composition. |
| 5 | 1–20 | Read the shared story in a familiar language. | Find the two-colour counting error. |
| 6 | Selected five | Open matching cards in a second language. | Keep the meaning stable while the expression changes. |
| 7 | Review | Explain C and D without looking at the answers. | Compare shares, then revisit just one uncertain word. |
If the child says “half each because two colours”, ask them to number the individual tokens R1, R2, R3 and B1. Which tokens make the red statement true? Three. Which make blue true? One. Remove the temporary numbers once the grouping is clear. This repairs the counting unit rather than asking the child to memorise an unexplained fraction.
If the difficulty is “blue must be next”, draw a line between the record and the bag. The record grows; the replaced contents remain three red and one blue. Ask what physically changed to make blue more likely. Under our independent model, nothing did. Do not demand a streak-breaking result to prove the point: any next colour is compatible with these chances.
If the learner explains correctly but hesitates over a translated term, keep the reasoning and practise only that language link. If the word is fluent but the explanation confuses prediction with certainty, return to one blue token in A. A vocabulary gap and a reasoning gap need different next steps; neither is a diagnosis of the child.
Look for an explanation that survives a new bag. C has four red and two blue tokens, all six individually equally likely. Red has 4 out of 6 chances and blue 2 out of 6; both are possible and red is more likely. D has one red and one blue: the colours are equally likely there. Having fewer tokens does not itself make a colour more likely; compare its share of the total.
A useful checkpoint is “I can tell the composition, describe an uncertain next outcome and report a frequency without rewriting either.” Invite the child to explain C in one language and point to the matching words in another. More attempts can supply more evidence, but a long record still does not guarantee the next result or excuse ignoring how the draws were made.
| Bag | Red tokens | Blue tokens | Red chance | Blue chance |
|---|---|---|---|---|
| C | 4 | 2 | 4/6 = 2/3 | 2/6 = 1/3 |
| D | 1 | 1 | 1/2 | 1/2 |
Does “random” mean every colour has equal chance?
No. An ideal random individual-token choice gives each token equal chance here, but a colour with more tokens groups more of those chances. Always name what is equally likely.
Does likely mean certain?
No. Red is likely in A but can fail on a particular draw. Certain means every allowed outcome satisfies the statement; red or blue is certain in this model.
Is a prediction wrong whenever the less likely outcome occurs?
A claim of certainty would be wrong. A well-qualified prediction of greater likelihood can remain sound even when the other outcome occurs. Check the wording, model and procedure rather than erasing the result.
Does fairness mean equal results?
Not in our agreed one-draw colour game. It means the two players have equal chances before the draw. Other games need their own fairness criteria, including rules and rewards; do not apply this one definition to every social situation.
Why replace the token?
To restore the stated composition for the next trial. Together with our independent ideal-draw assumption, this keeps the next colour chances unchanged. Replacement alone is not proof of good mixing in a real bag.
Will more trials guarantee an exact split?
No fixed finite run must match the model’s proportions exactly. More well-conducted trials can make frequencies useful evidence about the process, but do not guarantee a specified next colour. Investigate procedure problems rather than hiding inconvenient observations.
Must children learn the fractions first?
No. Start by pointing to the tokens and saying “three ways out of four”. Fractions can label that meaning when ready. Keep the equal-individual-chance assumption visible; memorising 3/4 without the setup is not the goal.
How can a child use all four languages without overload?
Keep one meaning and one small example stable. Explain it in the strongest language, open the same numbered card elsewhere and practise a short phrase. Five well-understood words are a useful session; finishing all editions is optional.
Background reading for adults
Adult background references on assigning chances and comparing models with experimental results. All token bags, records, word cards and stories here are original examples. The ideal equal-token model is stated explicitly; no real classroom results or national curriculum requirements are claimed.
- AMSI · Assigning probabilities to outcomes
- NRICH · Tricky Track: experimental and theoretical probability
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