Pratica di Shadowing: Quant Interview Questions #2 | Bayes Theorem - Impara a parlare inglese con i video
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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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Informazioni su questa lezione
In questa lezione, i partecipanti avranno l'opportunità di praticare la loro comprensione della probabilità, esplorando il teorema di Bayes attraverso un esempio pratico riguardante il lancio di monete. Gli studenti si eserciteranno nell'interpretazione di situazioni probabilistiche, apprendendo come calcolare le probabilità condizionate. Questo esercizio non solo migliorerà le abilità logiche, ma offrirà anche spunti utili per migliorare la pronuncia e la fluidità nella lingua inglese, utilizzando tecniche come il shadowspeak e il shadow speech.
Vocabolario e frasi chiave
- Coin - Moneta
- Probability - Probabilità
- Normal coin - Moneta normale
- Double-headed coin - Moneta doppia faccia
- Flip - Lanciare (una moneta)
- Event - Evento
- Outcome - Risultato
- Conditioned probability - Probabilità condizionata
Consigli per la pratica
Per massimizzare l'efficacia delle vostre sessioni di apprendimento attraverso il shadowspeaks, seguite questi suggerimenti specifici. Iniziate ascoltando attentamente la pronuncia e il tono del narratore nel video. Ripetete le frasi ad alta voce, cercando di imitare il ritmo e l'intonazione. Poiché il discorso è tecnico ma chiaro, provate a suddividere le frasi lunghe in parti più piccole per facilitarne la memorizzazione. Questo vi aiuterà a guadagnare sicurezza nell'uso di termini come "probabilità" e "risultati".
Oltre a praticare ripetendo le frasi, provate a utilizzare le parole chiave apprese in contesti diversi. Creare delle brevi storie o descrizioni in inglese utilizzando il nuovo lessico favorirà una migliore ritenzione e farà avanzare le vostre competenze linguistiche. Non dimenticate di utilizzare il shadowing site disponibile per ascoltare e praticare con ulteriori materiali audio che si affiancano a questa lezione.
La grammatica di questo video
Le strutture che chi parla usa di più, con le parole esatte del video:
| Struttura | Nel video |
|---|---|
| Present perfect have/has + participio passato — un’azione passata che conta ancora adesso | have flipped · have already calculated · You've answered |
| Forma passiva be + participio passato — conta ciò che accade, non chi lo fa | was drawn · is scaled · is represented |
| Frasi relative who / which + frase — un’informazione in più su una persona o una cosa | fraction, which is · this, which is · coin, which is |
Cos'è la tecnica dello Shadowing?
Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.