Binomial distributions | Probabilities of probabilities, part 1

Binomial distributions | Probabilities of probabilities, part 1

Part 2:
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These animations are largely made using manim, a scrappy open-source python library:

If you want to check it out, I feel compelled to warn you that it’s not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then “add subtitles/cc”. I really appreciate those who do this, as it helps make the lessons accessible to more people.


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  1. Pontus Tönder on April 14, 2020 at 3:01 pm

    Dude you gotta release the other parts soon, I got an exam in statistics coming up😂👌

  2. Somanath Dash on April 14, 2020 at 3:05 pm

    That means sucess rate can be find if we take more no. Of cases?

  3. Matt Paulson on April 14, 2020 at 3:06 pm

    This video kept teaching me things I had forgotten from statistics right before I could remember them

  4. Giuseppe Romagnuolo on April 14, 2020 at 3:08 pm

    I simply love all your videos, thank you!

  5. Josue Reynoso on April 14, 2020 at 3:11 pm

    This is a great example of the limitations of math models. The calculations are interesting and flawless, yet the model is clueless about how the world works (i.e. fake reviews) and thus useless in practice.

  6. Abraar Sameer on April 14, 2020 at 3:12 pm

    Can someone suggest a good, possibly almost complete book on probability theory for beginners?

  7. imsorrybutyournotgettingmyrealname sopleasesodoff on April 14, 2020 at 3:13 pm

    The other thing is it is easier for companies to write a solid but small amount of face positive reviews but it much more effort and less likely for them to that in a greater fashion.

  8. bttypst on April 14, 2020 at 3:13 pm

    some added skepticism for me with the seller’s with fewer ratings is the likelihood that those reviews are bots or biased in some way.

  9. Mohammad Mahdi on April 14, 2020 at 3:17 pm

    Could you provide a way to determine between item reviews which aren’t only 0/1 reviews and for example are 0 to 5 Stars? Thanks as always.

  10. Technoultimategaming on April 14, 2020 at 3:17 pm

    Imagine having 5 arrows and probability of shooting 1 is 5/7

    what is the probability that you hit exactly 5 times?


    5/7 • 5/7 • 5/7 • 5/7 • 5/7

    what if you have 6 arrows and want 5 of them to hit the target?

    Well. It’s 5 for successes and 1 failiure

    (5/7)^5 (2/7)

    Now was is the probability that it will hit on 1, 2, 3, 4, 5 or 6th arrow?

    there are 6 possibilities, 1 can fail, 2,3,4,5 or 6 can fail

    Out of 6 arrows we *choose* 1 arrow to be a fail


    and now expand the question to probability that 4 of them hit.

    2C6 are possibilities

    and in general we have



  11. Buck Starchaser on April 14, 2020 at 3:17 pm

    What is the probability of being suggested a multi-part lesson that starts at the beginning and flows continuously to the last part? I’m finding that it is so low, that it somewhat spoils the content.

  12. Gabriel Ribeiro on April 14, 2020 at 3:18 pm

    Heeey 3Blue1Brown,

    I kept a question in my mind.
    In a test in which every question has 5 possible answers and only one is correct. A normal multiple choice test. If 870 candidates answered A, 30 answered B, 20 answered C, 40 answered D and 40 answered E, can we infer that A is the correct answer?

    I enjoy your videos, thank you, bye.

  13. Andrew Montgomery on April 14, 2020 at 3:18 pm

    What about the probability of giving a review? People might be more likely to give a review if they had a negative experience. So your actual chance of having a good experience could be greater than what the data from the reviews would indicate. Or maybe it’s vice versa. Maybe you are more likely to give a review if you had a positive experience. Thus swinging the actual chance of a good experience lower than what the data would indicate. How would this be accounted for?

  14. Andrew Wilson on April 14, 2020 at 3:18 pm

    "someone with a true perfect success rate would never have those two negative reviews" — this is assuming that the negative reviews reflected a failure on the part of the provider, but people leave bad reviews all the time for reasons beyond the provider’s control.

    Furthermore, there’s an effect with online reviews where bad experience self-select to review the product or service more frequently than average experiences and extreme positive reviews tend to self-select more than average, but less than negative experiences. This generally gets into the problem of poor sampling and manipulation of the metric. Unless the review is mandatory the bimodal 1 and 5 star review distribution seems to reign. Then there are methods that are used by app developers to prompt first to see if the user had a positive experience, if so, ask them to review it, and if not, ask them to provide feedback (but crucially not link them to the reviews)

  15. Tor Lumnitor on April 14, 2020 at 3:20 pm

    what about the probability of the unreported experiences? That is the probability that someone with a bad experience will bother to write a review, or that a problem with a vehicle will be caught on the line. or visa versa.

  16. Untitled 1 on April 14, 2020 at 3:21 pm

    Can you please bring back the old title of:
    "Which review sample should you chose Mathematically speaking?" .
    We dont want to stumble on Mandela Effects.

  17. Jon Dewey on April 14, 2020 at 3:23 pm

    Great vid.
    Does NOT actually apply to Amazon!!!

    From Amazon’s website:
    //How does Amazon calculate star ratings?
    Amazon calculates a product’s star ratings based on a machine learned model instead of a raw data average. The model takes into account factors including the age of a rating, whether the ratings are from verified purchasers, and factors that establish reviewer trustworthiness.//

  18. Krystal Myth on April 14, 2020 at 3:23 pm

    You can also rule out negative reviews by reading them. Finding out if they are worth counting or not.

  19. JIM TSIO on April 14, 2020 at 3:24 pm

    I flip a coin and it comes heads! I have the evidence the coin will give heads 100% of the time!!!🤣🤣🤣

  20. Nazka231 on April 14, 2020 at 3:26 pm

    I just want the part 2 and 3 now

  21. Marcus Helmer on April 14, 2020 at 3:26 pm

    In my model of what’s happening, I tend to assume that there are a few extra reviews from friends, employees, or alternate/fake accounts that bolster the positivity of the experience. Especially if there aren’t a lot reviews.

  22. 5000 subs with no videos on April 14, 2020 at 3:27 pm

    What id instead of just positive or negative reviews, ratings were on a 0 to 1 scale, 0 being completely negative and 1 being completely positive

  23. Ralph Kang on April 14, 2020 at 3:28 pm

    Still waiting for part 2 hahaha

  24. Christoph G. on April 14, 2020 at 3:29 pm

    10:52 What’s the answear to the question I don’t get it why do the curves have different sizes. Is the area under the curve = 1.

  25. Eduard 23 on April 14, 2020 at 3:29 pm

    how do you calculate that give me a general formula

  26. Paul Johnson on April 14, 2020 at 3:29 pm

    How do you factor in the regular practice of sellers giving away free products for inflated reviews?

  27. Maibrl on April 14, 2020 at 3:30 pm

    I just want to thank you for the series, that’s exactly what I need to understand for my upcoming maths finals, at least part 1 and two afaik

  28. JD Zellers on April 14, 2020 at 3:30 pm

    When’s part 2 coming out??

  29. Maksym Karunos on April 14, 2020 at 3:31 pm

    Well, while the video is a hundred percent correct from the math precpective, Amazon reviews are a little bit more complicated that a raw average of reviews ratings. I know there is a whole team of ML engineers who work on the model that adjusts these numbers

  30. Gabriel Rios-Perez on April 14, 2020 at 3:32 pm

    "assuming reviews are independent"…ready to see where this goes

  31. Nevo Krien on April 14, 2020 at 3:33 pm

    Whats this therom called

  32. None Of That on April 14, 2020 at 3:34 pm

    I feel like your statement at 10:05 about "would happen one in a 1000 times" contradicts the whole point of this video, of probability vs data

  33. Ian Storey on April 14, 2020 at 3:37 pm

    Very nice

  34. Apple Juice on April 14, 2020 at 3:38 pm

    YouTube recommendations are weird

  35. ilker sar on April 14, 2020 at 3:38 pm

    my brain over, sorry, ypu keep going..

  36. earth trighton on April 14, 2020 at 3:38 pm

    Breath better, take a break from coughing

  37. s 1291 on April 14, 2020 at 3:42 pm

    You are making things more complex.

  38. gaurav mishra on April 14, 2020 at 3:43 pm

    plz do complex no series. i want it too badly

  39. 좀빈 on April 14, 2020 at 3:43 pm

    한글자막 감사합니다.

  40. John Snow on April 14, 2020 at 3:45 pm

    It’s amazing how probability can be both intuitive and unintuitive at different times. Like, humans tend to be good at intuitively understanding bayesian probability, but bad at intuiting other types of probability. I guess I shouldn’t be surprised though, since people are good at intuitively understanding certain types of math (like counting, addition, subtraction…etc.), and are completely baffled by others.

  41. ಠ_ಠ on April 14, 2020 at 3:46 pm

    I don’t think Kruger and Dunning took into account me seeing this video 🤯🤯🤯

  42. violentcat345 on April 14, 2020 at 3:46 pm

    Suggestion! I noticed your use of colours to link values, I imagine the colours you have used would not have the intended helpfulness for people with colour blindness, perhaps try using blue/orange or red/orange instead as most people with colour blindness can distinguish blue. Thank you for the fantastic video, looking forward to seeing the rest of the series!

  43. Pond on April 14, 2020 at 3:48 pm

    I made it incredibly simple… 200 is 3x more reviews than 50, 96% is 3% more than 93%, therefore, to me, they are exactly the same. I know this isn’t accurate at all, I should be good at maths, but I’m really not.

  44. Lee on April 14, 2020 at 3:50 pm

    Oh my goodness, these videos are so awesome. I am stunned. I have a degree in statistics and I love this.

  45. LeROYtheLAMA on April 14, 2020 at 3:50 pm

    I’m 100% that 99% watching are as smart as 98% of the population and 92% left about 65% into the video

  46. Aidan Fitzgerald on April 14, 2020 at 3:52 pm

    6:30 Monte Carlo go brrrr

  47. WooHyung Choi on April 14, 2020 at 3:52 pm

    자막 감사합니당

  48. A. P. V. on April 14, 2020 at 3:53 pm

    Didn’t understand anything, but it was very interesting.

  49. KC Sutherland on April 14, 2020 at 3:56 pm

    This video is about to help some people get jobs

  50. Thaplayer1209 The player on April 14, 2020 at 4:00 pm

    A bit late but can anyone tell me how to see which is best if there are more then 2 possible ratings? Like those 1-5 star reviews. Do I have to take an extra review of each possible rating and then take the average to find the "best" one