Pratique du Shadowing: Quant Interview Questions #2 | Bayes Theorem - Apprendre l'anglais à l'oral avec la vidéo
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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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Vocabulaire et conseils d’expression pour cette leçon
Cette leçon d’expression orale de niveau C1 s’appuie sur la vidéo « Quant Interview Questions ». Les mots qui reviennent le plus souvent : coin, probability, normal, headed, double. Cette vidéo contient 124 phrases et 1546 mots à répéter en shadowing. La partie parlée dure 10:02. Le locuteur parle à un rythme naturel d’environ 154 mots par minute, proche d’une conversation courante. Seuls 80 % des mots font partie des 3 000 mots les plus courants en anglais, le vocabulaire est donc exigeant. 22 de ses 124 phrases sont des questions : un bon entraînement pour l’intonation interrogative et les réponses courtes.
Vocabulaire clé de cette vidéo
Les 15 mots les plus avancés de la vidéo, avec leur prononciation et leur sens :
| Mot | Prononciation | Sens |
|---|---|---|
| probability nom | /ˌpɹɑ.bəˈbɪl.ə.ti/ | probabilité |
| posterior adjectif | /poʊˈstɪɹ.i.ɚ/ | postérieur |
| calculate verbe | /ˈkælkjʊleɪt/ | calculer |
| fraction nom | /ˈfɹæk.ʃən/ | fraction |
| mathematical adjectif | /ˌmæθ(.ə)ˈmæt.ɪ.kəl/ | mathématique |
| coefficient nom | /ˌkoʊ.ɪˈfɪʃənt/ | coefficient, facteur |
| intersect verbe | /ɪntɚˈsɛkt/ | intersecter, interséquer |
| multiply verbe | /ˈmʌltɪplaɪ/ | multiplier |
| normalize verbe | /ˈnɔː.mə.laɪz/ | normaliser |
| playlist nom | /ˈpleɪˌlɪst/ | playlist, liste de lecture |
| theorem nom | /ˈθiərəm/ | théorème |
| equation nom | /ɪˈkweɪ.ʒən/ | équation |
| hint nom | /hɪnt/ | indice |
| bucket nom | /ˈbʌkɪt/ | seau, chaudière |
| completeness nom | complétude |
Les verbes à particule que vous entendrez
| Mot | Sens |
|---|---|
| break down verbe | tomber en panne |
La grammaire de cette vidéo
Les structures que le locuteur utilise le plus, avec les mots exacts de la vidéo :
| Structure | Dans la vidéo |
|---|---|
| Present perfect have/has + participe passé — une action passée qui compte encore maintenant | have flipped · have already calculated · You've answered |
| Voix passive be + participe passé — l’accent est mis sur ce qui arrive, pas sur qui le fait | was drawn · is scaled · is represented |
| Propositions relatives who / which + proposition — une précision sur une personne ou une chose | fraction, which is · this, which is · coin, which is |
Prononciation à surveiller
Le locuteur utilise 5 contractions et formes réduites, comme doesn't, don't, you'll. Prononcez-les sous leur forme courte, telles que vous les entendez.
- Les sons « th »: mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, theorem /ˈθiərəm/
- Les sons « sh » et « zh »: fraction /ˈfɹæk.ʃən/, coefficient /ˌkoʊ.ɪˈfɪʃənt/, equation /ɪˈkweɪ.ʒən/, notation /noʊˈteɪʃən/, conditional /kənˈdɪʃ.ə.nəl/
- Mots longs — placez bien l’accent: probability /ˌpɹɑ.bəˈbɪl.ə.ti/, posterior /poʊˈstɪɹ.i.ɚ/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, coefficient /ˌkoʊ.ɪˈfɪʃənt/, conditional /kənˈdɪʃ.ə.nəl/
Les sons difficiles pour les francophones :
- /r/ anglais — langue recourbée, sans frotter la gorge: probability /ˌpɹɑ.bəˈbɪl.ə.ti/, posterior /poʊˈstɪɹ.i.ɚ/, fraction /ˈfɹæk.ʃən/, theorem /ˈθiərəm/, randomly /ˈɹændm̩li/
Comment s’entraîner avec cette vidéo
- Écoutez la vidéo en entier une fois sans parler et notez les mots que vous ne connaissez pas.
- Commencez à la vitesse 0,75×, répétez phrase par phrase, puis revenez à la vitesse normale quand cela devient facile.
- Enregistrez-vous et comparez avec l’original, en faisant attention à des mots comme probability, posterior, calculate.
Qu'est-ce que la technique du Shadowing ?
Le Shadowing est une technique d'apprentissage des langues fondée sur la science, développée à l'origine pour la formation des interprètes professionnels. Le principe est simple mais puissant : vous écoutez de l'anglais natif et le répétez immédiatement à voix haute — comme une ombre suivant le locuteur avec un décalage de 1 à 2 secondes. Les recherches montrent une amélioration significative de la précision de la prononciation, de l'intonation, du rythme, des liaisons, de la compréhension orale et de la fluidité.