쉐도잉 연습: Quant Interview Questions #2 | Bayes Theorem - 영상으로 영어 말하기 배우기

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You have a bucket of 500 coins.

499 on normal coins?

but one of them is a double -headed coin.

You draw a coin randomly for the bucket and then you flip it five times.

Question one.

If you picked a normal coin, so if you drew a normal coin, What is the probability of flipping 5 hints?

Question 2: What is the probability of flipping 5 heads?

in general, just from the situation.

And lastly, What is the probability of picking a normal coin having obtained 5 hits from flipping the coin that was drawn.

Now from this last question, you can tell that we are trying to calculate a probability

using or knowing some new piece of information in this case is the knowledge that we have flipped Five hits.

using this random coin.

Hence, this is a question regarding Bayes' theorem listed on the right.

So the probability of A of event A given event B is equal to the probability of event A, so known as the prior,

And this is scaled by some fraction, which is the probability of event B given A.

normalized or divided by the probability of event B.

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

The probability of event A on the left is equals to the prior estimate of this, which is just probability of event A,

And this is scaled by some you could think about it scaling coefficient.

And this is represented by the fraction.

Now.

At this point, I like to just write out all the probabilities

that I can just obtain just from the question description itself.

So, Here we have some Straightforward ones, the probability of picking a normal coin, there are 499 of them out of the 500.

Therefore, it's 499 over 500 and the probability of picking a double -headed coin, which is 1 out of 500.

Now focusing on a normal coin, what is the probability of having a heads or tails, right?

Because this is a normal coin, we only have two outcomes for when we flip one, a heads or tails.

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.

Whereas for the double -headed coin, because it is double -headed, the only outcome you can get is hits.

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.

Given that you have a double -headed coin, It's just 1 because you'll always get it.

And the probability of getting tails, having the double -headed coin, is obviously 0 because it's double -headed.

So let's address question one.

If we pick a normal coin, what is the probability of flipping 5 hits with this normal coin?

So we flip it once, twice, 4 times, 5 times, what is the probability of getting 5 hits?

Now because these are five flips, 5 independent flips, so each flip does not affect the other

we can say that the probability of having 5 hits given that we have a normal coin is just Half?

times half, times half, times half, five times.

1 /2 to the power of 5, which is just 1 /32, right?

Because these are five independent flips, each of them having, with each outcome having an equal opportunity of occurring.

And you can also say that the probability of getting 5 Tails will be the same thing, right?

Because the probability of getting a hit full of tails on a normal coin is the same.

And for the sake of completeness, let's also calculate the above but for a double -headed coin.

So the probability of getting 5 hits Assuming you have a double -headed coin, it's naturally just 1.

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.

So that answers the first question.

The probability of getting 5 /8 given that we have a normal coin is 1 /32.

Thank you.

Now for the second question.

What is the probability of flipping 5 hits?

So we know that we have flipped.

a coin five times, right?

This coin could either be a double headed coin or a normal coin so

what is the probability of us just getting 5 hits, regardless of what type of coin we have?

So we can say

And I hope you are following

that the probability of getting 5 hits is the probability of getting 5 hits with a normal coin.

plus the probability of getting 5 hits with a double -headed coin.

Breaking that down further with more mathematical notation, it is the probability of getting 5 hits,

It's probably the event of 5 hits intersected with the event of getting a normal coin,

plus the probability of The event of five hits intersected with the event of getting a double -headed coin.

And breaking this down further using the laws of conditional probability, we can see that these two terms here.

these two terms here break down into these two terms here, alright?

using this equation here.

So, Let's think about what we have, right?

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.

And similarly for the second term, And if you think about it, we already have all these probabilities.

This is what's listed above us.

hence it's plugging in the values, we should get this.

1 over 32 times 499 over 500 plus 1, times 10500, which gives us 0 .03318, right?

So this is the probability of getting 5 hints.

Okay.

Hence, Going back to our question, And going back to our last question,

One is the probability of picking a normal coin, having obtained 5 hits from flipping the drawn coin.

So expressing that in mathematical terms is the probability of getting A Nullo coin?

given that we have flipped 5 hits.

So using Bayes' theorem, we can break this down into the probability of getting a normal coin,

our prior multiplied by this scaling coefficient, this fraction.

So it's the probability of getting 5 hits given that we have a normal coin.

normalized by the probability of getting 5 hits in general. and These

three terms we have already calculated from

either the question itself based on the description or from parts one and two above so just plugging in the values

we will see that The answer for part 3 is 0 .9399.

Great.

You've answered the question.

Thank you.

At this point, the interviewer may now ask you, does this result make sense?

Why or why not?

So let's think about what we have so far.

So we started with a prior, and we started with a prior estimate of Flipping.

um, Sorry, we started with a prior of the probability of getting a normal coin here,

which is just 499 out of 500 because there are 499 coins normal coins out of 500 coins.

And then we drew a coin and we had some new information about this having flipped five hits, So let's think about it.

Now our posterior, estimate of probability of getting a normal coin, is 0 .9399.

Right.

And if we just express the prior in a decimal form, just to make it easier for us, What we have is...

The probability of getting a normal coin, the prior, is 0 .998.

And now what you will notice is that our posterior 9399 is lower than our prior.

0 .998.

100

Why?

101

Why has the posterior changed and lower it, lower slightly compared to our prior.

102

So let's think about what this new information is.

103

This new information that was given to us was that we obtained 5 hits.

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Right?

105

Yeah. And...

106

The fact that we obtain 5 heads, this New information.

107

actually supports the other outcome which is us having a double -headed coin.

108

because the probability of getting a...

109

of flipping 5 hits using a normal coin is so much rarer compared to that of a double headed coin,

110

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.

112

It's dropped a little bit, Why?

113

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.

117

It actually supports us having likely drawn a double -headed coin.

118

Hence, we see this very small drop in the from the prior to the posterior estimate.

119

Now, depending on the I guess the structure of the question,

120

you might actually see that this prior to posterior drop in probabilities or maybe increase, depending on the question, can actually be quite large.

121

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.

123

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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이 레슨에 대한 소개

이번 레슨에서는 베이즈 정리에 대한 질문을 통해 확률 개념을 학습하고, 동전 던지기의 결과를 바탕으로 확률을 계산하는 연습을 합니다. 확률을 이해하고 적용하는 과정은 영어 학습에도 도움이 되며, 영어로 사고하는 능력을 키울 수 있습니다. 이 영상을 시청하면서 확률과 통계에 대한 이해를 높이고, 영어 말하기 실력을 향상시키는 데 중점을 두세요.

핵심 어휘 및 구문

확률 (Probability): 사건이 일어날 가능성
정상 동전 (Normal Coin): 공정한 동전, 앞면과 뒷면이 있다
더블 헤드 동전 (Double-Headed Coin): 앞면만 있는 동전
사건 (Event): 확률적으로 발생하는 일
독립적 사건 (Independent Events): 한 사건이 다른 사건에 영향을 미치지 않는 경우
사후 확률 (Posterior Probability): 새로운 정보를 바탕으로 한 확률
사전 확률 (Prior Probability): 새로운 정보가 없을 때의 확률
동전 던지기 (Coin Flipping): 동전을 던져 결과를 얻는 행위

연습 팁

영상에서 나오는 내용을 더 효과적으로 습득하기 위해 영어 쉐도잉 기법을 활용해보세요. 영상의 속도에 맞춰, 화자의 음성과 억양을 따라서 말해보는 것이 중요한데, 이는 여러분이 영어의 자연스러운 리듬과 발음을 익히는 데 큰 도움이 됩니다. 특히, 통계나 확률과 같은 전문 어휘를 연습할 때는 반복적으로 발음하고, 문맥 속에서 적절하게 사용해 보세요.

또한, shadowspeak와 같은 기술을 활용하여, 의사소통의 맥락을 깊이 이해하고, 질문에 대한 답변을 할 때 더 자연스럽고 자신감을 가질 수 있습니다. 유튜브 영어 공부 영상은 각기 다른 회화 스타일을 접할 수 있는 좋은 방법이므로, 자주 시청하고 반복해서 연습하면 큰 도움이 됩니다.

이 영상의 문법

화자가 가장 많이 쓰는 문형을 영상 속 실제 표현과 함께 정리했습니다.

문형	영상 속 표현
현재완료 have/has + 과거분사 — 과거의 일이 지금도 관련이 있을 때	have flipped · have already calculated · You've answered
수동태 be + 과거분사 — 누가 하는지보다 무슨 일이 일어나는지에 초점	was drawn · is scaled · is represented
관계절 who / which + 절 — 사람이나 사물에 대한 추가 정보	fraction, which is · this, which is · coin, which is

쉐도잉이란? 영어 실력을 빠르게 키우는 과학적 방법

쉐도잉(Shadowing)은 원래 전문 통역사 훈련을 위해 개발된 언어 학습 기법으로, 다언어 학자인 Dr. Alexander Arguelles에 의해 대중화된 방법입니다. 핵심 원리는 간단하지만 매우 강력합니다: 원어민의 영어를 들으면서 1~2초의 짧은 지연으로 즉시 소리 내어 따라 말하는 것——마치 '그림자(shadow)'처럼 화자를 따라가는 것입니다. 문법 공부나 수동적인 청취와 달리, 쉐도잉은 뇌와 입 근육이 동시에 실시간으로 영어를 처리하고 재현하도록 훈련합니다. 연구에 따르면 이 방법은 발음 정확도, 억양, 리듬, 연음, 청취력, 말하기 유창성을 크게 향상시킵니다. IELTS 스피킹 준비와 자연스러운 영어 소통을 원하는 분들에게 특히 효과적입니다.

섀도잉 방법: 단계별 전체 가이드 읽기 →

쉐도잉 연습: Quant Interview Questions #2 | Bayes Theorem - 영상으로 영어 말하기 배우기

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이 레슨에 대한 소개

핵심 어휘 및 구문

연습 팁

이 영상의 문법

쉐도잉이란? 영어 실력을 빠르게 키우는 과학적 방법