Shadowing Practice: Quant Interview Questions #2 | Bayes Theorem - Learn English Speaking with Video
Choose a practice mode
Each mode opens a full-screen practice room.
125 sentences
If sentences are too short or too long, click Edit to adjust them.
1
You have a bucket of 500 coins.
2
499 on normal coins?
3
but one of them is a double -headed coin.
4
You draw a coin randomly for the bucket and then you flip it five times.
5
Question one.
6
If you picked a normal coin, so if you drew a normal coin, What is the probability of flipping 5 hints?
7
Question 2: What is the probability of flipping 5 heads?
8
in general, just from the situation.
9
And lastly, What is the probability of picking a normal coin having obtained 5 hits from flipping the coin that was drawn.
10
Now from this last question, you can tell that we are trying to calculate a probability
11
using or knowing some new piece of information in this case is the knowledge that we have flipped Five hits.
12
using this random coin.
13
Hence, this is a question regarding Bayes' theorem listed on the right.
14
So the probability of A of event A given event B is equal to the probability of event A, so known as the prior,
15
And this is scaled by some fraction, which is the probability of event B given A.
16
normalized or divided by the probability of event B.
17
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
18
The probability of event A on the left is equals to the prior estimate of this, which is just probability of event A,
19
And this is scaled by some you could think about it scaling coefficient.
20
And this is represented by the fraction.
21
Now.
22
At this point, I like to just write out all the probabilities
23
that I can just obtain just from the question description itself.
24
So, Here we have some Straightforward ones, the probability of picking a normal coin, there are 499 of them out of the 500.
25
Therefore, it's 499 over 500 and the probability of picking a double -headed coin, which is 1 out of 500.
26
Now focusing on a normal coin, what is the probability of having a heads or tails, right?
27
Because this is a normal coin, we only have two outcomes for when we flip one, a heads or tails.
28
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.
29
Whereas for the double -headed coin, because it is double -headed, the only outcome you can get is hits.
30
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.
31
Given that you have a double -headed coin, It's just 1 because you'll always get it.
32
And the probability of getting tails, having the double -headed coin, is obviously 0 because it's double -headed.
33
So let's address question one.
34
If we pick a normal coin, what is the probability of flipping 5 hits with this normal coin?
35
So we flip it once, twice, 4 times, 5 times, what is the probability of getting 5 hits?
36
Now because these are five flips, 5 independent flips, so each flip does not affect the other
37
we can say that the probability of having 5 hits given that we have a normal coin is just Half?
38
times half, times half, times half, five times.
39
1 /2 to the power of 5, which is just 1 /32, right?
40
Because these are five independent flips, each of them having, with each outcome having an equal opportunity of occurring.
41
And you can also say that the probability of getting 5 Tails will be the same thing, right?
42
Because the probability of getting a hit full of tails on a normal coin is the same.
43
And for the sake of completeness, let's also calculate the above but for a double -headed coin.
44
So the probability of getting 5 hits Assuming you have a double -headed coin, it's naturally just 1.
45
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.
46
So that answers the first question.
47
The probability of getting 5 /8 given that we have a normal coin is 1 /32.
48
Thank you.
49
Now for the second question.
50
What is the probability of flipping 5 hits?
51
So we know that we have flipped.
52
a coin five times, right?
53
This coin could either be a double headed coin or a normal coin so
54
what is the probability of us just getting 5 hits, regardless of what type of coin we have?
55
So we can say
56
And I hope you are following
57
that the probability of getting 5 hits is the probability of getting 5 hits with a normal coin.
58
plus the probability of getting 5 hits with a double -headed coin.
59
Breaking that down further with more mathematical notation, it is the probability of getting 5 hits,
60
It's probably the event of 5 hits intersected with the event of getting a normal coin,
61
plus the probability of The event of five hits intersected with the event of getting a double -headed coin.
62
And breaking this down further using the laws of conditional probability, we can see that these two terms here.
63
these two terms here break down into these two terms here, alright?
64
using this equation here.
65
So, Let's think about what we have, right?
66
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.
67
And similarly for the second term, And if you think about it, we already have all these probabilities.
68
This is what's listed above us.
69
hence it's plugging in the values, we should get this.
70
1 over 32 times 499 over 500 plus 1, times 10500, which gives us 0 .03318, right?
71
So this is the probability of getting 5 hints.
72
Okay.
73
Hence, Going back to our question, And going back to our last question,
74
One is the probability of picking a normal coin, having obtained 5 hits from flipping the drawn coin.
75
So expressing that in mathematical terms is the probability of getting A Nullo coin?
76
given that we have flipped 5 hits.
77
So using Bayes' theorem, we can break this down into the probability of getting a normal coin,
78
our prior multiplied by this scaling coefficient, this fraction.
79
So it's the probability of getting 5 hits given that we have a normal coin.
80
normalized by the probability of getting 5 hits in general. and These
81
three terms we have already calculated from
82
either the question itself based on the description or from parts one and two above so just plugging in the values
83
we will see that The answer for part 3 is 0 .9399.
84
Great.
85
You've answered the question.
86
Thank you.
87
At this point, the interviewer may now ask you, does this result make sense?
88
Why or why not?
89
So let's think about what we have so far.
90
So we started with a prior, and we started with a prior estimate of Flipping.
91
um, Sorry, we started with a prior of the probability of getting a normal coin here,
92
which is just 499 out of 500 because there are 499 coins normal coins out of 500 coins.
93
And then we drew a coin and we had some new information about this having flipped five hits, So let's think about it.
94
Now our posterior, estimate of probability of getting a normal coin, is 0 .9399.
95
Right.
96
And if we just express the prior in a decimal form, just to make it easier for us, What we have is...
97
The probability of getting a normal coin, the prior, is 0 .998.
98
And now what you will notice is that our posterior 9399 is lower than our prior.
99
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.
Download App
Everything you need to speak fluently
AI PronunciationScore every sentence
IPA PracticeMaster every sound
VocabularyBuild your word bank
Vocab GameLearn while playing
FOR YOU
Recommend Video
C13
200 sentences
CAP Theorem in System Design Interviews
Business EnglishInterview EnglishUniversity Students
C12
41 sentences
Mathematical Statistics and the Theory of Probability in Casinos
Business EnglishUniversity StudentsWorking Adults
C12
257 sentences
A/B Testing in Data Science Interviews by a Google Data Scientist | DataInterview
Business EnglishInterview EnglishUniversity Students
C12
110 sentences
Two Sum - Leetcode 1 - HashMap - Python
Daily ConversationUniversity StudentsWorking Adults
C11
108 sentences
Cắt bánh - bài toán gây đau đầu nhiều thế kỷ
Daily ConversationUniversity StudentsWorking Adults
C13
128 sentences
The Most Controversial Problem in Philosophy
IELTS PrepTOEFL PrepUniversity Students
C13
101 sentences
How Does a URL Shortener Work?
Business EnglishInterview EnglishUniversity Students
B21
405 sentences
Answering behavioral interview questions is shockingly uncomplicated
Interview EnglishUniversity StudentsWorking Adults
C13
190 sentences
Rates and Returns – Module 1 – Quantitative Methods – CFA® Level I 2026
Cambridge Exam PrepUniversity StudentsWorking Adults
C1217 sentences
10 Data Analyst Interview Questions and Answers - Senior Data Analyst Explains
Interview EnglishUniversity StudentsWorking Adults
C1182 sentences
The Uncertainty Principle: What Does It Mean, How Does It Work?
Daily ConversationUniversity StudentsWorking Adults
C12
105 sentences
Faculty interviews: how to answer questions about the fit to the department #interview #phd
Interview EnglishUniversity StudentsWorking Adults
About This Lesson
In this lesson, you will delve into the fascinating world of probability, specifically focusing on Bayes' Theorem through engaging examples. By exploring real-life scenarios such as flipping coins, you will learn how to calculate and understand conditional probabilities. This exercise not only strengthens your analytical skills but also serves as an excellent opportunity to practice your English speaking and comprehension skills. As you learn the concepts of probability, you will simultaneously improve your English pronunciation, making this session an interactive and beneficial experience.
Key Vocabulary & Phrases
- Probability: The measure of the likelihood that an event will occur.
- Coin flip: An action of tossing a coin to decide between two possible outcomes.
- Normal coin: A standard coin with two distinct sides (heads and tails).
- Double-headed coin: A coin that has two identical sides, both showing heads.
- Independent events: Events whose outcomes do not affect each other.
- Conditional probability: The probability of an event occurring given the occurrence of another event.
Practice Tips
To maximize your learning from this video, consider utilizing a shadowing app to enhance your English speaking practice. Listen closely to the speaker’s tone, pace, and pronunciation, which is particularly clear in this video. As you practice, try to shadow speak along with the content, mimicking the way the presenter articulates each term. This method can greatly contribute to your ability to improve English pronunciation. The speaker's clear and methodical delivery—ideal for learners—allows for focused repetition. By maintaining the same speed, you can reinforce your understanding of both the probability concepts and the English language. Remember to pause the video when necessary to ensure accurate reproduction of phrases. Engaging with content designed for learning, such as "learn English with YouTube," can be very effective. Take your time, and feel free to revisit sections that challenge you to further enhance your English speaking practice.
Grammar in this video
The structures the speaker uses most, with the exact words from the video:
| Structure | In the video |
|---|---|
| Present perfect have/has + past participle — a past action that still matters now | have flipped · have already calculated · You've answered |
| Passive voice be + past participle — the focus is on what happens, not who does it | was drawn · is scaled · is represented |
| Relative clauses who / which + clause — extra information about a person or thing | fraction, which is · this, which is · coin, which is |
What is the Shadowing Technique?
Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.