birthday.html (7842B)
1 <!doctype html> 2 <html lang="en"> 3 <head> 4 <meta charset="utf-8"> 5 <meta name="viewport" content="width=device-width, initial-scale=1"> 6 <meta name="color-scheme" content="light dark"> 7 <title>We’re really bad at exponentials</title> 8 <link rel="shortcut icon" href="data:,"> 9 <style>body{font-family:system-ui,sans-serif;margin:auto;max-width:40em;padding:0 .5em 90vh}</style> 10 </head> 11 <body> 12 <article> 13 <p><a href="../">← Back</a></p> 14 <header> 15 <hgroup> 16 <h1>We’re really bad at exponentials</h1> 17 <p>The birthday paradox, Gaussian soap bubbles, and general coincidences</p> 18 </hgroup> 19 </header> 20 <p>Some things confuse me. To be fair, I don’t think I’m 21 especially good at riddles, I always fall for the famous paradoxes, 22 and</p> 23 <h2 id="the-birthday-paradox">The Birthday Paradox</h2> 24 <p>Most of 25 us (some of us prefer to) are surprised to learn 26 that it only 23 people in a classroom This is pretty simple to calculate 27 with basic combinatorics—we have 28 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false" form="prefix">(</mo><mtext mathvariant="normal">duplicates</mtext><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false" form="prefix">(</mo><mtext mathvariant="normal">no duplicates</mtext><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">P(\text{duplicates}) = 1 - P(\text{no duplicates})</annotation></semantics></math>$:</p> 29 <p>$ P() = $$</p> 30 <p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mn>365</mn><mi>!</mi><mi>/</mi><mo stretchy="false" form="prefix">(</mo><mn>365</mn><mo>−</mo><mn>23</mn><mo stretchy="false" form="postfix">)</mo><mi>!</mi></mrow><msup><mn>365</mn><mn>23</mn></msup></mfrac><annotation encoding="application/x-tex"> 31 \frac{365! / (365 - 23)!}{365^23} 32 </annotation></semantics></math></p> 33 <h2 id="gaussian-soap-bubbles">Gaussian soap bubbles</h2> 34 <p>I remember chatting with my supervisor Søren on the second day of the 35 hackathon that inaugurated my master’s thesis. It was about soap 36 bubbles. He explained that I found it very counterintuitive,</p> 37 <p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="normal">Geom</mtext><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mo minsize="240%" maxsize="240%" stretchy="true" form="prefix">(</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msup><mo minsize="240%" maxsize="240%" stretchy="true" form="postfix">)</mo><mrow><mn>1</mn><mi>/</mi><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> 38 \text{Geom}(x) = \bigg( \prod_{i=1}^N x_i \bigg)^{1 / N} 39 </annotation></semantics></math></p> 40 <h3 id="people-are-gaussian-too">People are Gaussian, too</h3> 41 <p>Another consequence of this comes up when I think about people.</p> 42 <ol type="1"> 43 <li>People are multifaceted in that they have many different properties 44 in terms of their life</li> 45 <li>When you look at a <em>single</em> Gaussian dimension things are, of 46 course, but no single point is actually that close to the mode!</li> 47 </ol> 48 <p>If you project a multivariate Gaussian onto any one of its 49 dimensions, it is univariate Gaussian. This is equivalent to saying that 50 the <em>marginal</em> distribution of a multivariate Gaussian is a 51 univariate Gaussian.</p> 52 <p>This is generally true for most finite-variance distributions, 53 though. The part that is unique to Gaussians is that it is 54 <em>uniform</em>…angular directions?</p> 55 <h3 id="multivariate-gaussians-and-multiple-univariate-gaussians">Multivariate 56 Gaussians and multiple univariate Gaussians</h3> 57 <p>Wait, is this a question of whether one person is “unique”, because 58 they have so many properties that they are unlikely to overlap in all of 59 them with someone else? I don’t think so. I think this is about the 60 likelihood being extremely small that <em>all</em> of someone’s 61 properties are close to the mode, i.e., “average”. I might be a pretty 62 regular guy in many ways, but it would be a wild coincidence if I was 63 regular in <em>every</em> way. This is <em>less</em> the case when you 64 consider that people’s properties are <em>not</em>, in fact, 65 independent, since some specific traits correlate with other specific 66 traits. However, the general argument still holds.</p> 67 <h2 id="graph-theory-or-six-degrees-of-separation">Graph theory, or six 68 degrees of separation</h2> 69 <p>I’m always surprised to find out that some friend-of-a-friend went to 70 school with a seemingly-unrelated friend of 71 mine (<a href="https://improbability-principle.com/tales-of-strange-coincidences/">and 72 so is everyone else</a>). We comment on how small the world is, or 73 remark that Copenhagen is a small town. This is probably also reflective 74 of me underestimating how many <em>things</em> there are. There are so 75 many ways we can be connected, so many close or distant relationships we 76 can have to one another, yet when some kind of connection arises, I’m 77 amazed by how unlikely it seems! The world is huge. We’re social 78 creatures that seek out community, so each of us is connected, with a 79 couple degrees of separation, to a very large number of people.</p> 80 <p>Our lives are long, and varied, and our brains are excellent at 81 discerning patterns—even ones that aren’t there. And there are a lot of 82 us! Our brains are also adept in subtly but impactfully altering reality 83 - updating our memories, etc. - to fit the narratives and “coincidences” 84 it wants to see. We <em>love</em> spurious correlations.</p> 85 <p>something something connect to independence in the distribution 86 again, clusters in the graphs etc</p> 87 <p>Julia knowing one of Beka’s friends - we have lots of friends, and 88 our friends have lots of friends We hang out with people who travel, 89 study abroad, and study engineering, and so do our friends. Eventually, 90 the exponential always takes over.</p> 91 <p>However, the opposite also holds. The clustering that exists 92 <em>exacerbates</em> the correlations - so we end up meeting Having 93 never met someone with the same name as you, and then, when you do, 94 being shocked that their parents <em>also</em> got the name from some 95 novel - I mean, it’s a rare name! And any rare name is <em>extra</em> 96 likely to have common sources of inspiration. This is because we live in 97 a <a href="https://en.wikipedia.org/wiki/Small-world_network">small 98 world</a>, which means In reality, this refers to local communities, 99 friends of friends But the correlations we see also This also influences 100 what we perceive to be “random”—it’s never really random, it correlates 101 with who we are So we wouldn’t find it particularly remarkable if we had 102 a connection to a truly “random” person, i.e., a random sample from the 103 population of people on the planet. We find it remarkable when we have a 104 connection with a <em>specific</em> person, and the fact that we thought 105 of that person already implies a possible correlation with them</p> 106 <h2 id="combinatorics-and-exponentials">Combinatorics and 107 exponentials</h2> 108 <p>Behind all of these things is combinatorics. We just seem to be 109 really bad at making <a href="https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow">system 110 1</a> predictions that involve combinatorics.</p> 111 <p>In these cases, the combinatorial effect of adding more features 112 starts outweighing the effect of the Gaussian pulling features towards 113 the mode.</p> 114 <p>We’re terrible We think in logarithmic terms, so we consistently 115 underestimate the power of exponentials.</p> 116 <p>Climate change, and covid</p> 117 </article> 118 </body> 119 </html>