跟讀練習: Super Simple Explanation of Probability Tree Diagram! - 通過影片學習英語口語
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116 句
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Today we're going to learn a probability tree diagram.
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Imagine a spinner that has six equal sections.
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Two sections are colored blue, while the remaining four sections are colored red.
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A student named Jake spins this spinner twice.
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Our first goal is to simply list every possible outcome that can happen after the two spins.
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Since Jake spins the spinner two times, each spin has only two possible outcomes.
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The spinner can land on blue or red during the first spin, and once again on blue or red during the second spin.
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If we write every possible combination, we get four outcomes in total.
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These are blue followed by blue, blue followed by red, red followed by blue, and finally red followed by red.
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Writing all the outcomes like this is manageable for a small example, but as the number of events increases, the list quickly becomes long and confusing.
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This is exactly why we use a probability tree diagram.
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Instead of writing every possibility separately, we organize them into branches so that every possible outcome can be seen clearly.
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We always begin with a single starting point.
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Since the first spin has only two possible outcomes, we draw two branches coming out from this point.
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One branch represents blue, while the other branch represents red.
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These two branches account for every possible result of the first spin.
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Now we move to the second spin.
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No matter what happened on the first spin, the second spin can again result in either blue or red.
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Therefore, from each existing branch, we draw two more branches.
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One branch represents blue and the other represents red.
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If we now trace every complete path from left to right, we obtain exactly the same four outcomes we listed earlier.
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The first path represents blue, followed by blue.
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The second path represents blue, followed by red.
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The third path represents red, followed by blue.
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The final path represents red, followed by red.
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At this point, the tree diagram shows every possible outcome, but it does not yet include any probabilities.
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To add those, we first look at the spinner.
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Since two of the six sections are blue, the probability of landing on blue is 2 out of 6, which simplifies to 1 out of 3.
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The remaining four sections are red, so the probability of landing on red is four out of six, which simplifies to two out of 3.
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We now write these probabilities beside each branch of the tree.
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Every branch representing blue is labeled 1 out of 3, while every branch representing red is labeled 2 out of 3.
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Since the same spinner is used again for the second spin, these probabilities remain exactly the same on every second level branch.
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Our probability tree diagram is now complete.
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It shows every possible outcome together with the probability of taking each branch.
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Now let's look at another example where we'll use a probability tree diagram to calculate probabilities.
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A student named Emily is going to roll a fair six-sided die and then flip a fair coin.
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She wins a prize only if she rolls a four on the die and then gets heads on the coin toss.
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Our task is to complete the probability tree diagram.
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Okay, we begin by looking at the die.
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There are only two possible outcomes that matter for this question.
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Emily either rolls a four or she does not roll a four.
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Since there is only one face showing four on a fair six-sided die, the probability of rolling a four is one out of six.
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The remaining five faces are not four, so the probability of not rolling a four is five out of six.
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We write these probabilities on the first pair of branches.
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Together, they add up to one, as they always should.
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Now we move to the second stage of the tree diagram, which represents the coin toss.
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From each of the two branches that we have already drawn, we again draw two new branches.
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One branch represents heads, while the other represents tails.
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This is because after either rolling a four or not rolling a four, the coin can still land on either heads or tails.
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A fair coin has two equally likely outcomes, heads and tails.
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Therefore, the probability of getting heads is one out of two, and the probability of getting tails is also one out of two.
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We write these probabilities on both sets of second-level branches because the coin is fair regardless of the result of the die.
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Our probability tree diagram is now complete.
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The next step is to use this tree diagram to calculate the probability of different outcomes.
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Suppose we want to calculate the probability that Emily wins the prize.
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According to the question, she wins only if she first rolls a four and then gets heads.
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We begin by locating this root on the tree diagram.
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First, we follow the branch labeled 4, and then we follow the branch labeled heads.
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Whenever we want to find the probability of one complete root through a probability tree, we multiply the probabilities along that root.
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So in this example, we will multiply 1 by 6 and half.
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This is the fundamental rule that is used in every probability tree question.
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This gives us 1 by 12.
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Therefore, the probability that Emily wins the prize is 1 out of 12.
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Amazing!
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Now let's look at one last example involving probability tree diagrams.
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A bag contains 4 blue marbles and 6 yellow marbles.
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A student named Michael randomly picks one marble from the bag, replaces it, and then picks another marble.
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We need to complete the probability tree diagram and answer a few probability questions.
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Since there are two possible outcomes for the first pick, blue or yellow, we begin by drawing two branches from the starting point.
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Now we move to the second pick.
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Because the second pick can again result in either blue or yellow, we draw two more branches from each of the first branches.
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This gives us all the possible outcomes for the two picks.
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Now, the first step is to determine the probabilities for the first pick.
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Since there are four blue marbles and six yellow marbles, there are ten marbles altogether.
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Therefore, the probability of picking a blue marble is four out of 10,
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while the probability of picking a yellow marble is 6 out of 10.
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These probabilities are written on the first pair of branches.
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Then we write the probabilities for the second pick.
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Since the first marble is placed back into the bag before the next pick, the contents of the bag remain exactly the same.
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Therefore, the probability of picking a blue marble is still 4 out of 10,
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while the probability of picking a yellow marble is still 6 out of 10.
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We write these probabilities beside each of the second-level branches, and our probability tree diagram is now complete.
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Next, suppose we are asked to calculate the probability that both marbles selected are blue.
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We first identify the correct root on the tree diagram?
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This route follows the blue branch on the first pick and then the blue branch on the second pick.
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Since we are following one complete route, we multiply the probabilities along that path.
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The probability of blue followed by blue is 4 out of 10 multiplied by 4 out of 10.
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Therefore, the probability of selecting two blue marbles is 16 out of 100, which is 16%.
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Now here's how this question becomes more interesting and fun.
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Suppose we are asked to calculate the probability of selecting exactly one blue marble.
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There are two different ways this can happen.
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The first possibility is selecting blue on the first pick and yellow on the second pick.
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The second possibility is selecting yellow on the first pick and blue on the second pick.
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Let us calculate the probability for the first route.
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The probability of blue followed by yellow is 4 out of 10 multiplied by 6 out of 10, which is 24 out of 100.
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Now consider the second possible route.
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The probability of yellow followed by blue is 6 out of 10 multiplied by 4 out of 10, which is 24 out of 100.
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Since both of these routes satisfy the condition of selecting exactly one blue marble, we must add their probabilities together.
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24 by 100 plus 24 by 100 equals 48 out of 100.
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Notice that this time we are adding the probabilities because there is more than one successful route.
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Whenever multiple different paths satisfy the required condition, we add their probabilities together.
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Finally, suppose we are asked to calculate the probability that both selected marbles are the same color.
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There are again two successful routes.
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The first route is blue followed by blue, while the second route is yellow followed by yellow.
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Since both roots satisfy the condition, we calculate each one separately.
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We already know that the probability of blue followed by blue is 16 out of 100.
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The probability of yellow followed by yellow is 6 out of 10 multiplied by 6 out of 10
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or 36 out of 100.
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Since both roots produce marbles of the same color, we add their probabilities together.
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16 by 100 plus 36 by 100 equals 52 out of 100.
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This is the probability that both selected marbles have the same color.
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And that's how probability tree diagrams work.
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Just remember the two key rules.
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Multiply probabilities along a single path and add the probabilities of different successful paths.
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If you enjoyed this video, please don't forget to like, share, and subscribe to our channel.
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Also, you can support my channel by joining our community and becoming a member. So good!
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本課的詞彙與口語要點
這節 B1 級別的口語課以影片“Super Simple Explanation of Probability Tree Diagram!”為素材。 影片中反覆出現的詞有：probability, blue, marble, probabilities, tree。 這段影片共有 116 個句子、1569 個單詞可供跟讀。 講話部分時長為 11:29。 說話人語速平穩，每分鐘約 137 個詞，很適合跟讀。 82% 的單詞屬於英語最常用的 3,000 詞，其餘的詞建議在練習前先學一下。
影片中的重點詞彙
影片中最難的 15 個單詞，附發音和釋義：
| 單詞 | 發音 | 釋義 |
|---|---|---|
| probability 名詞 | /ˌpɹɑ.bəˈbɪl.ə.ti/ | 可能性 |
| marble 名詞 | /ˈmɑɹ.bəl/ | 大理石 |
| diagram 名詞 | /ˈdaɪ.ə.ɡɹæm/ | 圖 /圖, 圖表 /圖表 |
| multiply 動詞 | /ˈmʌltɪplaɪ/ | 增長 /增長 |
| calculate 動詞 | /ˈkælkjʊleɪt/ | 計算 /計算, 算 |
| coin 名詞 | /ˈkoɪ̯n/ | 硬幣 /硬幣, 銅錢 /銅錢 |
| satisfy 動詞 | /ˈsætɪsfaɪ/ | 滿足 /滿足 |
| simplify 動詞 | /ˈsɪmplɪfaɪ/ | 簡化 /簡化 |
| toss 動詞 | /tɔs/ | 拋 /拋 |
| separately 副詞 | /ˈsɛp.ɹət.li/ | 分別 /分別, 單獨 /單獨 |
| confuse 動詞 | /kənˈfjuːz/ | 讓……困惑 /讓……困惑 |
| randomly 副詞 | /ˈɹændm̩li/ | 隨機 |
| organize 動詞 | /ˈɔɹɡənaɪz/ | 組織 /組織 |
| subscribe 動詞 | /səbˈskɹaɪb/ | 簽署 /簽署 |
| trace 名詞 | /tɹeɪs/ | 痕跡 /痕跡 |
影片中的語法
說話人最常用的結構，並附上影片中的原話：
| 結構 | 影片中的用法 |
|---|---|
| 被動語態 be + 過去分詞 — 強調發生了什麼，而不是誰做的 | are colored · can be seen · is labeled |
| 現在進行時 am/is/are + -ing — 此刻或近期正在發生的事 | are following · is selecting · are adding |
| “going to” 表將來 be going to + 動詞 — 計劃或明顯即將發生的事 | we're going to learn · is going to roll |
| “will” 表將來 will + 動詞 — 決定、承諾或預測 | we'll use · will multiply |
需要注意的發音
說話人用了 3 次縮略和弱讀形式，例如 don't, we'll, we're。請按聽到的簡短形式來說。
- 長單詞——注意重音位置: probability /ˌpɹɑ.bəˈbɪl.ə.ti/, manageable /ˈmænəd͡ʒəb(ə)l/, altogether /ɔl.tuˈɡɛð.ɚ/
中文母語者容易讀錯的音：
- 詞尾子音 — 要讀清楚，後面不要加母音: calculate /ˈkælkjʊleɪt/, toss /tɔs/, beside /bɪˈsaɪd/, confuse /kənˈfjuːz/, locate /ˈloʊˌkeɪt/
- /r/ 和 /l/ 的區別: probability /ˌpɹɑ.bəˈbɪl.ə.ti/, marble /ˈmɑɹ.bəl/, separately /ˈsɛp.ɹət.li/, randomly /ˈɹændm̩li/
- /æ/ — 嘴張大，不要讀成 /e/: diagram /ˈdaɪ.ə.ɡɹæm/, calculate /ˈkælkjʊleɪt/, satisfy /ˈsætɪsfaɪ/, manageable /ˈmænəd͡ʒəb(ə)l/, randomly /ˈɹændm̩li/
如何用這段影片練習
- 先完整聽一遍影片，不要開口，記下不認識的單詞。
- 用正常速度逐句跟讀，每句重複到你的節奏與說話人一致為止。
- 錄下自己的聲音並與原聲對比，特別注意 probability, marble, diagram 這類單詞。
什麼是跟讀法？
跟讀法 (Shadowing) 是一種有科學依據的語言學習技巧，最初開發用於專業口譯員的培訓，並由多語言者Alexander Arguelles博士普及。這個方法簡單而強大：您在聽英語母語原聲的同時立即大聲重複——就像是一個延遲1-2秒緊跟說話者的影子。與被動聽力或語法練習不同，跟讀法強迫您的大腦和口腔肌肉同時處理並模模擬實的講話模式。研究表明它能顯著提高發音準確性，語調，節奏，連讀，聽力理解和口語流利度——使其成為雅思口語備考和真實英語交流最有效的方法之一。