シャドーイング練習: Quant Interview Questions #2 | Bayes Theorem - 動画で英語スピーキングを学ぶ

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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.

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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.

124

And don't forget to like and subscribe.

125

Thank you.

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このレッスンについて

このレッスンでは、ベイズの定理に関連する確率の問題について学びます。具体的には、500枚のコインの中からコインを1枚ランダムに引き、そのコインを5回投げた時の結果に基づいてさまざまな確率を計算します。この学習を通じて、確率の基礎を理解し、英語での思考を深めることができます。この内容は、YouTubeで英語学習をしている方や、IELTSスピーキング対策をしたい方に特に有益です。

重要な語彙とフレーズ

確率 (probability)
コイン (coin)
引く (draw)
投げる (flip)
正常なコイン (normal coin)
ダブルヘッダーコイン (double-headed coin)
アウトカム (outcome)
事前確率 (prior probability)

練習のコツ

このビデオでは、確率というテーマに沿ったスピードとトーンで話されているので、shadow speakを活用して練習することをお勧めします。特に、このビデオの内容に合わせてシャドーイングを行い、各文のリズムやイントネーションを意識しながら声に出すことが重要です。最初はゆっくりとしたペースから始め、慣れてきたらスピードを上げると良いでしょう。また、重要なフレーズや語彙を繰り返し口にすることで、自然な表現力や発音を向上させることができます。この方法は、IELTSスピーキング対策にも非常に役立ちますので、積極的に取り入れてみてください。

この動画の文法

話し手がよく使っている文型を、動画の実際の表現とともに紹介します。

文型	動画での表現
現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している	have flipped · have already calculated · You've answered
受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる	was drawn · is scaled · is represented
関係詞節 who / which + 節 — 人や物について情報を加える	fraction, which is · this, which is · coin, which is

シャドーイングとは？英語上達に効果的な理由

シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。

シャドーイングのやり方: ステップ別の完全ガイドを読む →

シャドーイング練習: Quant Interview Questions #2 | Bayes Theorem - 動画で英語スピーキングを学ぶ

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重要な語彙とフレーズ

練習のコツ

この動画の文法

シャドーイングとは？英語上達に効果的な理由