Answers.Body: <p>From a transcript at <a href="https://www.springfieldspringfield.co.uk/view_episode_scripts.php?tv-show=mr-robot-2015&amp;episode=s02e04" rel="nofollow noreferrer">https://www.springfieldspringfield.co.uk/view_episode_scripts.php?tv-show=mr-robot-2015&amp;episode=s02e04</a></p>
  
  <blockquote>
    <p>You just need to be able to see ahead of your opponent. </p>
    
    <p>But how can
    you when your other opponent is you? <strong>Stalemate</strong>.</p>
    
    <p>Three in a row.</p>
    
    <p>The
    odds of that are Impossible.</p>
  </blockquote>
  
  <p>So the impossibility mentioned can be interpreted to be three stalemates in a row, not simply three draws.  I could not quickly find a reference to the stalemate rate, but it must be significantly less than the draw rate, and the cube of it will therefore tend a bit closer to "impossible".  </p>
  
  <p>I don't know if the moves in the show are meant to be real games or just a nod to the audience that is looking closely (kind of like the in-jokes written in tiny print on "Star Trek" displays).  But the first two are <a href="https://www.chess.com/forum/view/general/chess-games-in-tv-show-mr-robot" rel="nofollow noreferrer">reported</a> to be the <a href="https://en.wikipedia.org/wiki/Stalemate#In_problems" rel="nofollow noreferrer">Loyd 10-move stalemate</a> and the final game is Minerva's fastest known <a href="https://en.wikipedia.org/wiki/Stalemate#Double_stalemate" rel="nofollow noreferrer">double stalemate</a>.   (I don't have a copy of the video to confirm this)</p>
  
  <p>Because they are "constructed" puzzles, neither would be expected in a competitive game, only a cooperation scenario.  That may or may not mean something in the context of the episode.  </p>
  
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Answers.UserDisplayName: BowlOfRed
Answers.Body: <p>As standard math says, probability of <em>same</em> event happening in a row for <code>N</code> times is: </p>
  
  <blockquote>
    <p>P<sub>e</sub><sup>N</sup></p>
  </blockquote>
  
  <p>where P<sub>e</sub> is single event probability.</p>
  
  <p>So if we take as standard draw probability 50%, then event of having 3 draws in a row between some players will be 1/2<sup>3</sup> = 0,125 = <code>12.5%</code>. Thus not impossible, but <strong>unlikely</strong>.</p>
  
  <p>However there can be specific pairs of players which always plays at optimal strategy for each-other, so that their games mostly ends in a draw.</p>
  
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Answers.Text: 12.5% isn't "highly unlikely" at all. It means that, if you played three games a day, you'd expect to get three draws almost once a week. Also, the 50% draw rate is in long-format games between grandmasters; quicker games between amateurs have a much lower rate of draws.
Answers.UserDisplayName: David Richerby
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Answers.Text: @DavidRicherby. I've changed to simply `unlikely`, but in principle it's just a play of words. I simply had in mind that statistically probability of **NOT** getting a draw 3-times in a row is `87.5%`, so this is much more likely. You can't argue with mathematics
Answers.UserDisplayName: Agnius Vasiliauskas
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Answers.Text: It's not "a play of words": it's a significant change of meaning. I can't argue with the fact that, if three independent events occur with probability p, then all three occur together with probability p^3. But I can (and did!) argue with your choice of p=1/2, which does not reflect reality, and the implicit claim of independence, though the latter probably isn't significant in this case.
Answers.UserDisplayName: David Richerby
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Answers.Text: I never said that 50% is "carved in a rock", this was just as an example  - have you noticed word 'if' ? And yes - I know that these events must be independent for formula to apply. I can't image dependent events in this case - unless with each start of new game players gets more drunk :-D BTW, 50Į-60% is not also for grandmasters and between computer chess programs too.
Answers.UserDisplayName: Agnius Vasiliauskas
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Answers.UserDisplayName: Agnius Vasiliauskas
Answers.Body: <p>Definitely not impossible! It is true that draws are less frequent in amateur chess, and it is also true that draws are less frequent for faster games (how fast was the game in the show?), but even so, let's say that for a blitz game between evenly matched amateurs the draw frequency were 5%, or 1/20. If you played three games, the odds of playing three draws would be 1/8000, which while unlikely, is hardly "impossible". This is assuming that the probabilities are independent, which may not be entirely true but I think is close enough for a rough estimate.</p>
  
  <p>I don't have an academic citation, but if you want some stats for online games you can look at the Lichess opening book: <a href="https://lichess.org/analysis" rel="noreferrer">https://lichess.org/analysis</a> . Normally it shows master-level games, but if you click on the settings you can chose to look at Lichess games instead, and filter by rating and time control. For blitz games at rating 1600, the draw frequency is 4-5%, depending on the opening move (the draw is the small gray stripe without a percentage, so you have to compute it by difference):</p>
  
  <p><a href="https://i.sstatic.net/BnTFx.png" rel="noreferrer"><img src="https://i.sstatic.net/BnTFx.png" alt="enter image description here"></a></p>
  
  <p>For the highest ratings, it's closer to 10% for blitz. And for classical chess, as you know, it's about 50% for top-level chess. For amateur chess I'd say maybe 10-20% but it depends on which level of amateur we are talking about. I won't cite the Lichess stats here because even though they have a "classical" category, it's really what would called rapid in over-the-board tournaments (30 min games in Lichess; OTB classical is usually 2 hours or more). </p>
  
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Answers.Text: lol this is exactly what I was going to do. Have an upvote!
Answers.UserDisplayName: Brandon_J
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Answers.Text: The recent TCEC superfinal between the Stockfish and Leela engines had 81 draws of 100 games, with openings handpicked to give one side a moderate advantage. It seems the top engines are approaching 100% draws
Answers.UserDisplayName: JollyJoker
Answers.ContentLicense: CC BY-SA 4.0
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Answers.Text: The games were crazy fast, or at least filmed in a way that suggests them playing bullet or Hyper-Bullet (30s).
Answers.UserDisplayName: Benjamin Raabe
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Answers.Text: I got some more statistics from lichess for the 30-60s + 0 time controls. Draw rate (Draw + Stalemate) is around 3%. :) https://imgur.com/a/RnNI1Sy
Answers.UserDisplayName: Benjamin Raabe
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Body: <p>In Mr. Robot (TV series) episode "eps2.2_init_1.asec", Elliot and Mr. Robot drew 3 consecutive chess games. They remark it's impossible.</p>
  
  <blockquote>
    <p>The odds of that are Impossible.</p>
  </blockquote>
  
  <p>As far as TV shows go, Mr. Robot is known for its technical accuracy. I was surprised when I heard that line, so I fact-checked it:</p>
  
  <p>From <a href="https://en.wikipedia.org/wiki/Draw_(chess)#Frequency_of_draws" rel="noreferrer">Wikipedia</a>:</p>
  
  <blockquote>
    <p>[Frequency of draws] is currently "holding pretty steady around 50%, and is only increasing at a very slow rate".</p>
  </blockquote>
  
  <p>I understand that figure is for top-level tournaments, not 3 consecutive games from the same 2 players. Is there are any accuracy to that statement in the show? If not, is it different from the 50% rate for top-level players?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-02-25T03:38:53.067Z
Comments.Text: Just to be clear: Mr. Robot is a person, and not some superhuman android or something?
Comments.UserDisplayName: D M
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-02-25T08:36:03.580Z
Comments.Text: Both Elliot and Mr. Robot are humans / computer geniuses / hackers. There is no indication that they are more than amateurs in chess.
Comments.UserDisplayName: Federico Poloni
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-02-25T09:00:15.813Z
Comments.Text: I just read online, that Elliot actually STALEMATED Mr. Robot three times - I will re-watch and confirm that episode after work, I only recall the games being... a little suspicious. ;) If that would be the case however, Elliot's remark may be understood as talking about the odds of him making the same mistake multiple times.
Comments.UserDisplayName: Benjamin Raabe
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-02-25T09:19:04.223Z
Comments.Text: @BenjaminRaabe "suspicious" is an understatement.  The games are replays of well-known constructions of stalemate.  While the moves are legal, you'd never see those games played in a competitive situation.
Comments.UserDisplayName: BowlOfRed
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-02-25T15:34:09.243Z
Comments.Text: @BowlOfRed: Why don't you answer the question then? If the answer is that it is not just any old draws, but actually impossible draws under competitive play, then that's the answer.
Comments.UserDisplayName: user21820
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-02-25T16:55:14.797Z
Comments.Text: @BowlOfRed: But the question is: "Is there [] any accuracy to **that statement in the show**?" and so the answer is "Yes because you misunderstood it." or "Meaningless question because it was staged and not like a random event.".
Comments.UserDisplayName: user21820
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Comments.CreationDate: 2019-02-25T17:32:01.057Z
Comments.Text: @BenjaminRaabe: thank you for bringing that point to my attention, it's been so long since I played chess it didn't cross my mind.
Comments.UserDisplayName: anon
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Comments.CreationDate: 2019-09-28T23:42:12.627Z
Comments.Text: Pretty low. Assuming the chance of a draw is 33% (among amateur players), it's 0.33^3
Comments.UserDisplayName: Inertial Ignorance
Tags: <draw><statistics>
Title: What is the draw frequency for 3 consecutive games (same players; amateur level)?
ThreadText: # What is the draw frequency for 3 consecutive games (same players; amateur level)?
  
  (Asked by: username_0 on 2019-02-25)
  
  In Mr. Robot (TV series) episode "eps2.2\_init\_1.asec", Elliot and Mr. Robot drew 3 consecutive chess games. They remark it's impossible.
  
  > The odds of that are Impossible.
  
  As far as TV shows go, Mr. Robot is known for its technical accuracy. I was surprised when I heard that line, so I fact-checked it:
  
  From [Wikipedia](https://en.wikipedia.org/wiki/Draw_(chess)#Frequency_of_draws):
  
  > [Frequency of draws] is currently "holding pretty steady around 50%, and is only increasing at a very slow rate".
  
  I understand that figure is for top-level tournaments, not 3 consecutive games from the same 2 players. Is there are any accuracy to that statement in the show? If not, is it different from the 50% rate for top-level players?
  
  **Tags:** draw, statistics
  
  > **Comments:**
  > **username_1**: Just to be clear: Mr. Robot is a person, and not some superhuman android or something?  
  > **username_2**: Both Elliot and Mr. Robot are humans / computer geniuses / hackers. There is no indication that they are more than amateurs in chess.  
  > **username_3**: I just read online, that Elliot actually STALEMATEusername_1r. Robot three times - I will re-watch and confirm that episode after work, I only recall the games being... a little suspicious. ;) If that would be the case however, Elliot's remark may be understood as talking about the odds of him making the same mistake multiple times.  
  > **username_4**: @BenjaminRaabe "suspicious" is an understatement.  The games are replays of well-known constructions of stalemate.  While the moves are legal, you'd never see those games played in a competitive situation.  
  > **username_5**: @username_4: Why don't you answer the question then? If the answer is that it is not just any old draws, but actually impossible draws under competitive play, then that's the answer.  
  > **username_5**: @username_4: But the question is: "Is there [] any accuracy to **that statement in the show**?" and so the answer is "Yes because you misunderstood it." or "Meaningless question because it was staged and not like a random event.".  
  > **username_0**: @BenjaminRaabe: thank you for bringing that point to my attention, it's been so long since I played chess it didn't cross my mind.  
  > **username_6**: Pretty low. Assuming the chance of a draw is 33% (among amateur players), it's 0.33^3  
  
  ---
  
  ## 3 answers
  
  ### Answer by username_9 on 2019-02-25. Score: 9 | Accepted answer
  
  Definitely not impossible! It is true that draws are less frequent in amateur chess, and it is also true that draws are less frequent for faster games (how fast was the game in the show?), but even so, let's say that for a blitz game between evenly matched amateurs the draw frequency were 5%, or 1/20. If you played three games, the odds of playing three draws would be 1/8000, which while unlikely, is hardly "impossible". This is assuming that the probabilities are independent, which may not be entirely true but I think is close enough for a rough estimate.
  
  I don't have an academic citation, but if you want some stats for online games you can look at the Lichess opening book: <https://lichess.org/analysis> . Normally it shows master-level games, but if you click on the settings you can chose to look at Lichess games instead, and filter by rating and time control. For blitz games at rating 1600, the draw frequency is 4-5%, depending on the opening move (the draw is the small gray stripe without a percentage, so you have to compute it by difference):
  
  [![enter image description here](https://i.sstatic.net/BnTFx.png)](https://i.sstatic.net/BnTFx.png)
  
  For the highest ratings, it's closer to 10% for blitz. And for classical chess, as you know, it's about 50% for top-level chess. For amateur chess I'd say maybe 10-20% but it depends on which level of amateur we are talking about. I won't cite the Lichess stats here because even though they have a "classical" category, it's really what would called rapid in over-the-board tournaments (30 min games in Lichess; OTB classical is usually 2 hours or more).
  
  > **Comments:**
  > **username_10**: lol this is exactly what I was going to do. Have an upvote!  
  > **username_11**: The recent TCEC superfinal between the Stockfish and Leela engines had 81 draws of 100 games, with openings handpicked to give one side a moderate advantage. It seems the top engines are approaching 100% draws  
  > **username_3**: The games were crazy fast, or at least filmed in a way that suggests them playing bullet or Hyper-Bullet (30s).  
  > **username_3**: I got some more statistics from lichess for the 30-60s + 0 time controls. Draw rate (Draw + Stalemate) is around 3%. :) https://imgur.com/a/RnNI1Sy  
  
  ---
  
  ### Answer by username_4 on 2019-02-25. Score: 3
  
  From a transcript at <https://www.springfieldspringfield.co.uk/view_episode_scripts.php?tv-show=mr-robot-2015&episode=s02e04>
  
  > You just need to be able to see ahead of your opponent.
  >
  > But how can
  > you when your other opponent is you? **Stalemate**.
  >
  > Three in a row.
  >
  > The
  > odds of that are Impossible.
  
  So the impossibility mentioned can be interpreted to be three stalemates in a row, not simply three draws. I could not quickly find a reference to the stalemate rate, but it must be significantly less than the draw rate, and the cube of it will therefore tend a bit closer to "impossible".
  
  I don't know if the moves in the show are meant to be real games or just a nod to the audience that is looking closely (kind of like the in-jokes written in tiny print on "Star Trek" displays). But the first two are [reported](https://www.chess.com/forum/view/general/chess-games-in-tv-show-mr-robot) to be the [Loyd 10-move stalemate](https://en.wikipedia.org/wiki/Stalemate#In_problems) and the final game is Minerva's fastest known [double stalemate](https://en.wikipedia.org/wiki/Stalemate#Double_stalemate). (I don't have a copy of the video to confirm this)
  
  Because they are "constructed" puzzles, neither would be expected in a competitive game, only a cooperation scenario. That may or may not mean something in the context of the episode.
  
  
  ---
  
  ### Answer by username_7 on 2019-02-25. Score: 2
  
  As standard math says, probability of *same* event happening in a row for `N` times is:
  
  > PeN
  
  where Pe is single event probability.
  
  So if we take as standard draw probability 50%, then event of having 3 draws in a row between some players will be 1/23 = 0,125 = `12.5%`. Thus not impossible, but **unlikely**.
  
  However there can be specific pairs of players which always plays at optimal strategy for each-other, so that their games mostly ends in a draw.
  
  > **Comments:**
  > **username_8**: 12.5% isn't "highly unlikely" at all. It means that, if you played three games a day, you'd expect to get three draws almost once a week. Also, the 50% draw rate is in long-format games between grandmasters; quicker games between amateurs have a much lower rate of draws.  
  > **username_7**: @DavidRicherby. I've changed to simply `unlikely`, but in principle it's just a play of words. I simply had in mind that statistically probability of **NOT** getting a draw 3-times in a row is `87.5%`, so this is much more likely. You can't argue with mathematics  
  > **username_8**: It's not "a play of words": it's a significant change of meaning. I can't argue with the fact that, if three independent events occur with probability p, then all three occur together with probability p^3. But I can (and did!) argue with your choice of p=1/2, which does not reflect reality, and the implicit claim of independence, though the latter probably isn't significant in this case.  
  > **username_7**: I never said that 50% is "carved in a rock", this was just as an example  - have you noticed word 'if' ? And yes - I know that these events must be independent for formula to apply. I can't image dependent events in this case - unless with each start of new game players gets more drunk :-D BTW, 50Į-60% is not also for grandmasters and between computer chess programs too.  
  
  ---
  
  