跟讀練習: Quant Interview Questions #2 | Bayes Theorem - 通過影片學習英語口語

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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.

104

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.

111

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?

114

Because Overwhelmingly, most of the coins are still normal coins.

115

Therefore, it's still likely that we have drawn a normal coin.

116

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

122

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.

124

And don't forget to like and subscribe.

125

Thank you.

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為什麼要通過這個影片練習口語？

通過觀看和模仿這個關於貝葉斯定理的量化面試問題的影片，您可以提高自己的英語口語能力。雅思口語練習的關鍵在於理解不同的語法結構和表達方式，同時能夠在實際對話中靈活運用。這個影片通過生動的例項講解機率論，幫助學習者在數學和語言的結合上建立更加清晰的理解。通過這種方式，您不僅可以掌握相關的英語表達，還能在實際場景中應用，提高口語流利度。

語法與表達結構分析

在影片中，演講者使用了一些重要的語法和表達結構。以下是幾點值得注意的內容：

條件句構造：演講者經常使用“如果...那麼...”的結構來說明機率問題，例如：“如果您選擇了一個正常的硬幣，翻轉五次的機率是多少？”這種結構在日常對話中也很常見，有助於清晰表達假設與結果之間的關係。
機率表達：使用“機率為...”這樣的表述來描述可能性，比如說：“從500個硬幣中選擇一個正常硬幣的機率是499/500。”這增強了用數字表達複雜想法的能力。
歸納與總結：影片中的總結部分，例如“所以這回答了第一個問題”，展示瞭如何將複雜資訊簡化和整合，適用於任何討論和報告的場合。

常見發音陷阱

在影片中，有一些單詞和短語可能會讓學習者在發音時感到困惑。例如：

“機率” (probability)：注意這個詞中的重音位置，初學者可能會錯誤地強調非重音音節。
“選擇” (pick)：在快速對話中，可能會聽到“pick”的發音變得含糊不清，因此應當練習清晰的發音。
數字與分數：如“499/500”，在複述時需要確保每個數字都發音準確，以避免誤解。

通過短時間的英語影子跟讀 (shadowspeaks) 練習，您可以有效提高英語發音 (提高英語發音)，更好地掌握專業術語和表達方式。此外，定期進行這樣的練習，不僅能提高您的口語能力，還能讓您在面對複雜的對話主題時顯得更加自信。

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
現在完成時 have/has + 過去分詞 — 過去發生但與現在仍有關聯的事	have flipped · have already calculated · You've answered
被動語態 be + 過去分詞 — 強調發生了什麼，而不是誰做的	was drawn · is scaled · is represented
定語從句 who / which + 從句 — 補充說明人或事物	fraction, which is · this, which is · coin, which is

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。

影子跟讀法: 閱讀完整分步指南 →

跟讀練習: Quant Interview Questions #2 | Bayes Theorem - 通過影片學習英語口語

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為什麼要通過這個影片練習口語？

語法與表達結構分析

常見發音陷阱

影片中的語法

什麼是跟讀法？