Luyện nói tiếng Anh bằng Shadowing qua video: Quant Interview Questions #2 | Bayes Theorem
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125 câu
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You have a bucket of 500 coins.
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499 on normal coins?
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but one of them is a double -headed coin.
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You draw a coin randomly for the bucket and then you flip it five times.
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Question one.
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If you picked a normal coin, so if you drew a normal coin, What is the probability of flipping 5 hints?
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Question 2: What is the probability of flipping 5 heads?
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in general, just from the situation.
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And lastly, What is the probability of picking a normal coin having obtained 5 hits from flipping the coin that was drawn.
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Now from this last question, you can tell that we are trying to calculate a probability
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using or knowing some new piece of information in this case is the knowledge that we have flipped Five hits.
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using this random coin.
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Hence, this is a question regarding Bayes' theorem listed on the right.
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So the probability of A of event A given event B is equal to the probability of event A, so known as the prior,
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And this is scaled by some fraction, which is the probability of event B given A.
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normalized or divided by the probability of event B.
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So, To break down this equation, we can think of it as the posterior, so our posterior estimate of probabilities, In this case, the posterior estimate of
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The probability of event A on the left is equals to the prior estimate of this, which is just probability of event A,
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And this is scaled by some you could think about it scaling coefficient.
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And this is represented by the fraction.
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Now.
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At this point, I like to just write out all the probabilities
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that I can just obtain just from the question description itself.
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So, Here we have some Straightforward ones, the probability of picking a normal coin, there are 499 of them out of the 500.
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Therefore, it's 499 over 500 and the probability of picking a double -headed coin, which is 1 out of 500.
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Now focusing on a normal coin, what is the probability of having a heads or tails, right?
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Because this is a normal coin, we only have two outcomes for when we flip one, a heads or tails.
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and you will agree that it is half and half for both of them, because it's a fair coin, a fair coin or normal coin in this case.
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Whereas for the double -headed coin, because it is double -headed, the only outcome you can get is hits.
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Therefore, the probability of drawing a hit or flipping a hit using a double -headed coin, or in mathematical speaks, the probability of Flipping ahead.
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Given that you have a double -headed coin, It's just 1 because you'll always get it.
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And the probability of getting tails, having the double -headed coin, is obviously 0 because it's double -headed.
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So let's address question one.
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If we pick a normal coin, what is the probability of flipping 5 hits with this normal coin?
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So we flip it once, twice, 4 times, 5 times, what is the probability of getting 5 hits?
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Now because these are five flips, 5 independent flips, so each flip does not affect the other
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we can say that the probability of having 5 hits given that we have a normal coin is just Half?
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times half, times half, times half, five times.
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1 /2 to the power of 5, which is just 1 /32, right?
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Because these are five independent flips, each of them having, with each outcome having an equal opportunity of occurring.
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And you can also say that the probability of getting 5 Tails will be the same thing, right?
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Because the probability of getting a hit full of tails on a normal coin is the same.
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And for the sake of completeness, let's also calculate the above but for a double -headed coin.
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So the probability of getting 5 hits Assuming you have a double -headed coin, it's naturally just 1.
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because you're always going to get ahead with a double -headed coin, therefore it's 1 to the power of 5, which is just 1.
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So that answers the first question.
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The probability of getting 5 /8 given that we have a normal coin is 1 /32.
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Thank you.
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Now for the second question.
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What is the probability of flipping 5 hits?
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So we know that we have flipped.
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a coin five times, right?
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This coin could either be a double headed coin or a normal coin so
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what is the probability of us just getting 5 hits, regardless of what type of coin we have?
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So we can say
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And I hope you are following
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that the probability of getting 5 hits is the probability of getting 5 hits with a normal coin.
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plus the probability of getting 5 hits with a double -headed coin.
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Breaking that down further with more mathematical notation, it is the probability of getting 5 hits,
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It's probably the event of 5 hits intersected with the event of getting a normal coin,
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plus the probability of The event of five hits intersected with the event of getting a double -headed coin.
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And breaking this down further using the laws of conditional probability, we can see that these two terms here.
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these two terms here break down into these two terms here, alright?
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using this equation here.
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So, Let's think about what we have, right?
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On the left, Here, we have the probability of getting 5 hits, given we have a normal coin, multiply by the probability of having a normal coin.
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And similarly for the second term, And if you think about it, we already have all these probabilities.
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This is what's listed above us.
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hence it's plugging in the values, we should get this.
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1 over 32 times 499 over 500 plus 1, times 10500, which gives us 0 .03318, right?
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So this is the probability of getting 5 hints.
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Okay.
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Hence, Going back to our question, And going back to our last question,
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One is the probability of picking a normal coin, having obtained 5 hits from flipping the drawn coin.
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So expressing that in mathematical terms is the probability of getting A Nullo coin?
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given that we have flipped 5 hits.
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So using Bayes' theorem, we can break this down into the probability of getting a normal coin,
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our prior multiplied by this scaling coefficient, this fraction.
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So it's the probability of getting 5 hits given that we have a normal coin.
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normalized by the probability of getting 5 hits in general. and These
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three terms we have already calculated from
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either the question itself based on the description or from parts one and two above so just plugging in the values
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we will see that The answer for part 3 is 0 .9399.
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Great.
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You've answered the question.
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Thank you.
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At this point, the interviewer may now ask you, does this result make sense?
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Why or why not?
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So let's think about what we have so far.
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So we started with a prior, and we started with a prior estimate of Flipping.
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um, Sorry, we started with a prior of the probability of getting a normal coin here,
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which is just 499 out of 500 because there are 499 coins normal coins out of 500 coins.
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And then we drew a coin and we had some new information about this having flipped five hits, So let's think about it.
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Now our posterior, estimate of probability of getting a normal coin, is 0 .9399.
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Right.
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And if we just express the prior in a decimal form, just to make it easier for us, What we have is...
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The probability of getting a normal coin, the prior, is 0 .998.
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And now what you will notice is that our posterior 9399 is lower than our prior.
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0 .998.
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Why?
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Why has the posterior changed and lower it, lower slightly compared to our prior.
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So let's think about what this new information is.
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This new information that was given to us was that we obtained 5 hits.
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Right?
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Yeah. And...
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The fact that we obtain 5 heads, this New information.
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actually supports the other outcome which is us having a double -headed coin.
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because the probability of getting a...
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of flipping 5 hits using a normal coin is so much rarer compared to that of a double headed coin,
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that this new information that we have is actually supporting the fact that we might actually have a double -headed coin.
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Therefore, this Um, posterior estimate of the probability of getting a normal coin has dropped.
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It's dropped a little bit, Why?
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Why did they drop a little bit?
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Because Overwhelmingly, most of the coins are still normal coins.
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Therefore, it's still likely that we have drawn a normal coin.
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but it's dropped a little bit because we know that this new evidence doesn't support us having drawn a normal coin.
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It actually supports us having likely drawn a double -headed coin.
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Hence, we see this very small drop in the from the prior to the posterior estimate.
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Now, depending on the I guess the structure of the question,
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you might actually see that this prior to posterior drop in probabilities or maybe increase, depending on the question, can actually be quite large.
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And if you want such a question, please refer to The other base question that's found on this playlist on the left
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And if you would like to learn more about my recent experience and also my mistakes from my recent quantum interviews, applying in London in the year 2020.
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3. please refer to the playlist on the right.
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And don't forget to like and subscribe.
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Thank you.
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Giới thiệu bài học này
Bài học này sẽ giúp người học hiểu rõ hơn về Định lý Bayes thông qua một ví dụ thú vị liên quan đến việc rút đồng tiền và xác suất. Chúng ta sẽ không chỉ thực hành việc tính toán xác suất mà còn cải thiện kỹ năng nghe và nói tiếng Anh thông qua phương pháp học shadow speech. Qua bài học này, bạn sẽ có cơ hội luyện tập cách diễn đạt ý tưởng và thuật ngữ liên quan đến xác suất, từ đó nâng cao khả năng nói tiếng Anh của mình.
Từ vựng & Cụm từ chính
- Xác suất (Probability)
- Đồng xu bình thường (Normal coin)
- Đồng xu hai mặt (Double-headed coin)
- Lật đồng xu (Flip the coin)
- Kết quả (Outcome)
- Ngẫu nhiên (Random)
- Thí nghiệm độc lập (Independent trials)
- Ước lượng hậu nghiệm (Posterior estimate)
Lời khuyên luyện tập
Khi theo dõi video, hãy cố gắng áp dụng phương pháp shadow speak bằng cách lặp lại những gì bạn nghe ngay sau khi người nói phát âm. Video có tốc độ vừa phải, vì vậy bạn có thể dễ dàng theo kịp và sao chép cách phát âm, ngữ điệu và nhịp điệu. Hãy chú ý đến các đoạn mà người nói giải thích rõ ràng về các thuật ngữ và quy trình xác suất, vì đây là những phần rất có ích cho việc luyện nghe nói qua video. Bạn cũng có thể ghi lại giọng nói của mình và so sánh với video, giúp bạn nhận ra được những điểm cần cải thiện trong cách phát âm và ngữ điệu của bản thân.
Đừng ngại thử nghiệm nhiều lần với các đoạn khác nhau trong video. Việc lặp lại này như một bài tập shadowing không chỉ giúp bạn ghi nhớ từ vựng mà còn nâng cao khả năng ứng dụng trong thực tế. Hãy chăm chỉ luyện tập với phần mềm shadowing nếu có cơ hội, để thuận lợi hơn trong việc cải thiện kỹ năng nói tiếng Anh của mình.
Ngữ pháp trong video
Những cấu trúc người nói dùng nhiều nhất, kèm đúng cụm từ trong video:
| Cấu trúc | Trong video |
|---|---|
| Thì hiện tại hoàn thành have/has + quá khứ phân từ — việc đã xảy ra nhưng còn liên quan đến hiện tại | have flipped · have already calculated · You've answered |
| Câu bị động be + quá khứ phân từ — nhấn vào việc xảy ra, không phải người làm | was drawn · is scaled · is represented |
| Mệnh đề quan hệ who / which + mệnh đề — thêm thông tin về người hoặc vật | fraction, which is · this, which is · coin, which is |
Phương Pháp Shadowing Là Gì?
Shadowing là kỹ thuật học ngôn ngữ có cơ sở khoa học, ban đầu được phát triển cho chương trình đào tạo phiên dịch viên chuyên nghiệp và được phổ biến rộng rãi bởi nhà đa ngôn ngữ học Dr. Alexander Arguelles. Nguyên lý cốt lõi đơn giản nhưng cực kỳ hiệu quả: bạn nghe tiếng Anh của người bản xứ và lặp lại to ngay lập tức — như một "cái bóng" (shadow) đuổi theo người nói với độ trễ chỉ 1–2 giây. Khác với luyện ngữ pháp hay học từ vựng bị động, Shadowing buộc não bộ và cơ miệng phải đồng thời xử lý và tái tạo ngôn ngữ thực tế. Các nghiên cứu khoa học xác nhận phương pháp này cải thiện đáng kể phát âm, ngữ điệu, nhịp điệu, nối âm, kỹ năng nghe và độ lưu loát khi nói — đặc biệt hiệu quả cho người luyện IELTS Speaking và muốn giao tiếp tiếng Anh tự nhiên như người bản ngữ.