#probability
@wolleysegap @jhavok @antonioserrata @athyHans @lednabwm @Willow What is more, there is nothing more #Selfish than the #SelfishWhispers to #ImaginaryFriends to bend the laws of #Physics and #Probability to grant #SelfishDesires.
@KikoKate @antonioserrata @lednabwm Hey, if someone already #beLIEve s in a #SkyDaddy a #ZombieJew and whispers #SelfishWhispers to #ImaginaryFriends to bend the laws of #Physics and #Probability to grant their #SelfishDesires, it's trivial to beLIEve Trump was fomenting an #Insurrection for the good of the country.

Here's how I use probability distributions to effectively and efficiently test fixes that address test flakiness. If you use other approaches to test such fixes, then please share them via replies.
https://rvprasad.medium.com/testing-fixes-for-test-flakiness-9e45e92db28f
Bit of a #random #walk this morning.Tossed a coin at every junction.Head left Tail right. Resulted in a 1024/1 meeting with an old workmate. #probability


Protecting The Texas Power Grid From Tropical Cyclones - Increasing Resilience By Protecting Critical Lines
--
https://arxiv.org/abs/2301.13793 <-- shared technical article
--
#GIS #spatial #mapping #ERCOT #Texas #infrastructure #powersupply #powergrid #powergrids #electricnetwork #extremeweather #climatechange #cyclone #tropicalcyclones #mitigation #risk #hazard #electricalsecurity #energysupply #energysecurity #poweroutages #poweroutage #USA #globalwarming #model #modeling #probability #probabilistic #wind #failure #transmissionlines #transmissionlines #hurricanes #supplyfailures #supplyshortages #outages
SEP:ssä Alan Hájek on uusinut entryään todennäköisyyden tulkinnoista, https://plato.stanford.edu/entries/probability-interpret/
ja Eric Schwitzgebel uskomuksen käsitteestä, https://plato.stanford.edu/entries/belief/. Suomeksihan tuo kääntyy huonosti, ja usein jopa pelkkä 'ajatus' taitaa vastata englannin 'beliefiä'.
Kind of like the ergodic hypothesis: the viral load due to exhaled air in a hallway with people passing through is equivalent to the viral load of a similar density of people standing around in the hallway. But need to correct for air currents and greater or less exhalation from walking vs talking. #probability #Epidemiology
I think today, we've all learnt an important lesson about built-in organisational redundancy and slack capacity, with regards to mitigating against rare probability but high risk events...

#StuartKauffman - Can #Science Provide #Ultimate #Answers?
https://www.youtube.com/watch?v=uhB-riHuxVs
#Philosophy #PhilosophyOfScience #PhilosophyOfBiology #Science #Physics #Biology #Evolution #UseFunction #Utility #Probability #Determinacy #Emergence #Biosphere #Life #Predictability #LawsOfPhysics #LawsOfNature #CloserToTruth #RobertKuhn
Every time I see comments that should seem outrageous, I have to remind myself that millions of people #BELIEVE in an #Omniscient #Omnipresent #Omnipotent #Skydaddy a #ZombieJew and #Whisper #SelfishWhispers to #ImaginaryFriends to bend the laws of #Physics and #Probability to grant their #SelfishDesires--and it's been happening for thousands of years, and they pay 10% of what they earn to keep their #SelfDelusion going.


@rbreich Every time I see these comments that should seem outrageous, I have to remind myself that millions of people #BELIEVE in an #Omniscient #Omnipresent #Omnipotent #Skydaddy a #ZombieJew and #Whisper #SelfishWhispers to #ImaginaryFriends to bend the laws of #Physics and #Probability to grant their #SelfishDesires--and it's been happening for thousands of years, and they pay 10% of what they earn to keep their #SelfDelusion going.
#MenasKafatos - #ManyWorlds of #QuantumTheory
https://www.youtube.com/watch?v=lvf4hZXJX-Q
#Philosophy #PhilosophyOfScience #Science #Physics #QM #QuantumMechanics #QuantumPhysics #HughEverett #Everett #Probability #WaveFunction #WaveFunctionCollapse #Reality #Realism #Metaphysics #Entanglement #NielsBohr #CloserToTruth #RobertKuhn
Coin tosses aren’t random after all.
#math #probability #statistics #rng #prng #notsoprng
https://boingboing.net/2023/10/10/coin-toss-not-so-random-after-all-says-groundbreaking-study.html
:lobster: Dear Friends I #assume,
As we #probability know, the chances of being right are #exponentially #reduced through #wolfenstein and other extreme #measures. How so?
- #Burning out our fatigue
- Ferences but not Con
- Directed Coms not only for commie #comics
- #Knot hating while #tied with lost thunking
- Never assume, intuitive without reason
#Beginners are getting #older, #odder and #om
niverous. To put it another whey. Free milk school … yes it is a thing …
The other day someone had left out a box labeled free food. The power of Father X-mas Papá Noel compels you!
#AlfredMele - #FreeWill: #Essence and #Nature
https://www.youtube.com/watch?v=xht-jg6NAAE
#Philosophy #PhilosophyOfFreeWill #Will #Causation #Determinism #Compatibilism #Libertarianism #Randomness #Probability #Consciousness #Awareness #Self #Choice #Possibility #Dualism #Soul #Mind #CloserToTruth #RobertKuhn
Charcoal #drawing inspired by Cournot's principle. The (contested) principle is named after Augustin Cournot, who wrote in his 1843 treatise on #probability and chance: "On regarde comme physiquement impossible qu’un cône pesant se tienne en équilibre sur sa pointe" (It is considered physically impossible for a heavy cone to balance on its tip) https://books.google.be/books?id=6By2ku7FBSgC&ots=jfjjAsOKFo&pg=PA77

Did you know math can serve justice? Probability theory is sometimes used in legal cases to evaluate the likelihood of certain events, helping courts make more informed decisions. ⚖️🎲 #Probability #LegalSystems
#DavidAlbert - Setting #Time Aright
https://www.youtube.com/watch?v=UHcsE3rZgDM
#Philosophy #PhilosophyOfScience #Science #Physics #Causality #Entropy #SecondLaw #2ndLaw #Thermodynamics #TimeReversalSymmetry #Symmetry #Asymmetry #Motion #EquationsOfMotion #Boltzmann #Probability #CloserToTruth #RobertKuhn
#KnowledgeByte:A #Statistical #Distribution, or #Probability distribution, describes how values are distributed for a field.
A brief introduction to Standard Statistical Distributions. These are commonly used in #MachineLearning.
https://knowledgezone.co.in/posts/Statistical-Data-Distributions-62bbf309335afaa5baf181ac

Here's an excerpt from my presentation on #GaussianSplatting in context, starting from scratch to remind everyone what Gaussian distributions look like. ;)
Rotating the covariance matrix of a 3D Gaussian and sampling points from it each frame.
#Statistics #Probability #GaussianSplatting #GaussianProcesses

Playing with a 2D Gaussian.
#Statistics #Probability #GaussianSplatting #GaussianProcesses
How do I solve special #relativity problems? #Probability problems? #Logic problems?
Answer: I use #diagrams. I absolutely can't solve these things in my head. For one thing, they are often so counterintuitive. I therefore always need to draw the appropriate diagram. That was my trick for doing well in these topics at #undergraduate level and I highly recommend it.

撲克牌遊戲對戰
如AlohaGO 人工智慧
#blackjack 勝率計算與影響因素,使用亂數模擬
第一集:https://sendvid.com/789memeu
第二集:https://sendvid.com/ing0z6iv
第三集:https://sendvid.com/6lkyk5d5
王老師的人工智慧技術
#博弈 #casino #poker #AI #alphago #probability #statistics #統計學 #統計 #機率 #機率論 #程式設計 #程式 #亂數 #模擬 #人工智能 #人工智慧 #課程 #知識 #科技 #技術 #教學 #娛樂 #遊戲 #撲克牌 #撲克牌遊戲 #21點

#Probability and #Statistics are at the foundation of all things #STEM. Yet, most #IT practitioners do their best to avoid having to study these subjects and content themselves by invoking APIs in Python.
One way to remedy this ill is to force them to study quantum mechanics. But good luck with that.
A better way to teach them how to deal with uncertainty might be to force them to use stochastic compilers that yield correct compiled code in accordance with some domain-specific probability distributions.
Um wait; every piece of software that the IT industry produces is already "stochastically correct" and, as such, this lot is already proficient in coping with uncertainty. So, yeah, never mind. My bad.
The "Just One More" Paradox – Kelly Criterion [video]
https://www.youtube.com/watch?v=_FuuYSM7yOo
#ycombinator #Kelly_Criterion #Mathematics #Investment #Stocks #Probability #Statistics #Coin_flip #Game #Paradox #Ergodicity #Ergodic #Median #3Blue_1Brown
Just heard on Dutch Radio 2: interview with Eric-Jan Wagenmakers. With his team, he tested the theoretical prediction (by Diaconis et al.) that coins are slightly likelier to land heads when they start the toss heads up: with a probability of 51% rather than 50%. They found strong support for it! 😁 IgNobel prize in the making?
Link to the preprint of the study: "Fair coins tend to land on the same side they started: Evidence from 350,757 flips"
https://arxiv.org/abs/2310.04153
#statistics #probability
The #math behind big numbers is quite fun! Want to play a fun #probability #game? Virtual Rock, Paper Scissors via YouTube, to see who's one-in-a-million!
" #Christian Has NO RESPONSE When I Explain #Miracle #Testimony "
https://www.youtube.com/watch?v=r1D9j_ETYl0
#christianity #superstition #magic #atheism #atheist #science #possibility #probability
#PeterVanInwagen - Does a #FineTuned #Universe Lead to #God?
https://www.youtube.com/watch?v=pGwRBNCbDZk&ab_channel=CloserToTruth
#Philosophy #PhilosophyOfScience #Science #PhilosophyOfReligion #Religion #PhilosophyOfCosmology #Cosmology #Logic #FineTuning #Probability #LawsOfPhysics #LawsOfNature #AnthropicPrinciple #Teleology #Multiverse #CloserToTruth #RobertKuhn
#GregoryChaitin - Is #Information #Fundamental?
https://www.youtube.com/watch?v=ByJe3abjDZo&ab_channel=CloserToTruth
#Philosophy #Science #PhilosophyOfScience #Metaphysics #Idealism #Matter #Probability #QM #QuantumMechanics #SchrodingerEquation #Numbers #Math #Maths #Mathematics #Computers #Computation #Universe #Theory #CloserToTruth #RobertKuhn
Nice work by @turion integrating live Bayesian learning into a Functional Reactive Programming app. That's the power of embedded probabilistic programming languages like Monad-Bayes:
Planning a course, "Chance and Contingency in Human Life" combining philosophy of probability (etc.) with philosophical applications re personal decision making, public policy, justice, inequality, etc. I have material, but looking for suggestions of additional readings, especially for the social/personal topics. As a starting point for phil prob, I may use Hacking's _An Introduction to Probability and Inductive Logic_, but open to suggestions. #philosophy #philosophyofscience #probability
I wonder if the same applies to 2-up?
"Fair coins tend to land on the same side they started: Evidence from 350,757 Flips"
https://arxiv.org/abs/2310.04153
#Mathematics #Maths #Math #Probability #Statistics #Stats #Coin #CoinToss #Sport #Sports #Game #Games

Here's a starter conversation for your #stats class today:
"Fair coins tend to land on the same side they started: Evidence from 350,757 Flips"
https://arxiv.org/abs/2310.04153
#math #statistics #probability
#StuartKauffman - What Does a #FineTuned #Universe Mean?
https://www.youtube.com/watch?v=aO2o11KUAJk&ab_channel=CloserToTruth
#Philosophy #PhilosophyOfCosmology #Cosmology #PhilosophyOfScience #Science #FineTuning #Complexity #ComplexityTheory #Probability #Evolution #Biology #PhaseTransition #PhaseTransitions #Multiverse #StringTheory #AdjacentPossible #CloserToTruth #RobertKuhn
But... Do you keep winning tickets in stock, or are you out? 🤔 #statistics #probability

#DavidWallace - #Philosophy of #Physics and #Cosmology
https://www.youtube.com/watch?v=bsW_p7gbK-c&ab_channel=CloserToTruth
#PhilosophyOfPhysics #PhilosophyOfCosmology #PhilosophyOfScience #QM #QuantumMechanics #SpaceTime #LawsOfNature #LawsOfPhysics #ScientificMethod #MeasurementProblem #MeasureProblem #Probability #Emergence #Observer #Consciousness #CloserToTruth #RobertKuhn
"A Frequentist and a Bayesian go to a bar ..."
My follow-up blogpost after last week's "A note on observing zero successes", now focusing on what goes on when n=1. See https://allthingsdatascience.blogspot.com/2023/09/a-frequentist-and-bayesian-go-to-bar.html. Enjoy!
#DataScience #Bayes #Bayesian #Statistics #Probability #MRX #MarketResearch

Just updated my Collection of Dice Problems with a problem suggested by a reader (it's #27 of the now 78 problems). I figure when I get to 100 problems I'll throw a party. https://www.madandmoonly.com/doctormatt/mathematics/dice1.pdf
In an unplanned exercise in #probability, I let my students of my Field Professionalism/Wilderness First Aid class this semester pick either of two weekends (whichever one works out better for them), and it's exactly 50-50 so far on which one they have picked. #math
Derivation of the correlation coefficient for a two channel Bell-test #experiment:
https://crudfactory.com/eprb_signal_correlations.pdf
#Simulation employing only action by contact (no ‘#entanglement’):
https://github.com/chemoelectric/eprb_signal_correlations
The derivation is done with #probability theory, employing no #quantum #physics whatsoever. It is shown that Bell, Clauser, et al., compute correlation coefficients incorrectly.
An animation is on PyPI. See another pinned toot.
(What #QuantumComputing really ‘is’ is a subject for future research.)
Sierpinski Fractal using Chaos Game: https://youtube.com/shorts/0qti3cwWHsI
#pico8 #youtube #youtuber #youtubeshorts #youtubeshort #pixel #pixelart #pixelartist #fractal #fractalart #fractals #shorts #short #programming #programmer #code #coding #creative #creativity #creativecoding #2d #2dart #math #mathart #mathematics #maths #chaos #probability #viral #trending
Derivation of the correlation coefficient for a two channel Bell-test experiment: https://github.com/chemoelectric/eprb_signal_correlations
The derivation is done with #probability theory, employing no #quantum #physics whatsoever. It is shown that quantum theorists attempting to analyze the problem using probability theory have computed expectations incorrectly, thus arriving at the wrong function.
A #simulation demonstrates that the solution here is correct.
Mirrors:
https://eprb-signal-correlations.sf.net
https://bitbucket.org/chemoelectric/eprb_signal_correlations/
I have a proof of the correlation coefficient for a two channel Bell-test experiment here, along with an #Ada simulation of such an experiment:
https://github.com/chemoelectric/eprb_signal_correlations
#Quantum #physics #QuantumComputing #Simulation #Signal #Processing #Electrical #Engineering #Probability
why is there no #statstadon for #statistics and #probability #math and #mathematics nerds? Or #risk #analysts, #actuaries , etc
@georgetakei Sadly, I am reminded that if you already #BELIEVE in an #Omnipotent #Omniscient #Omnipresent #SkyDaddy and #Whisper #SelfishWhispers to a #ZombieJew to bend the laws of #Physics and #Probability to grant your #SelfishDesires you can even believe in ANYTHING, even Trump
Nice write-up from Ole Peters about ergodicity and the "infamous" coin toss. I've been sharing these basic ideas in my math classes for the past few years, and I think it really introduces students to new and important ways of thinking.
https://ergodicityeconomics.com/2023/07/28/the-infamous-coin-toss/
(Re-posting this with a slight modification)
Exploring the law of probability, it suggests that we can calculate the likelihood of an uncertain event based on the known outcomes of several distinct events.
Here's a light-hearted scenario for your consideration:
You're on a walk and encounter a puddle perfectly centered on your path. What would your reaction be?
New LabPlot tutorial
https://tube.kockatoo.org/w/9Kmqeefqw35EpEZj914N5p
In this short video you'll learn how to how to fit a distribution to data.
#DistributionFitting #GaussianDistribution, #Log-normalDistribution #Probability Distribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood
This is what I got from #midjourney for the #Bayes Theorem.
No clue why the model came up with that but it's beautiful
#Math #Probability
@threkk A nice way to think about this problem is to imagine the version with 100 doors instead of 3.
You pick a door. The host then opens 98 of the remaining 99 doors revealing no prize, and then offers you the chance to switch to the one remaining door. Since switching wins if the prize was behind any of those 99 doors, it seems like a good idea!
This still sounds weird, and there is a long, interesting article on the Wikipedia about it, including the controversies. It is important to notice that the experimental simulations indeed confirm that switching is still the winning move. #probability #math https://en.wikipedia.org/wiki/Monty_Hall_problem?useskin=vector
However, after picking one, the host will show the content of one of the wrong doors, and ask you if you want to switch. Intuitively, you would think that switching or not does not matter, because it is still 50% chance... but it does: switching will increase your chances.
The key point is that the host will open one door that does not win. The fact that it is not random changes the probability, as the combined probability is higher on the remaining one. #probability #math
I was watching D.P. today when they talked about the Monty Hall problem. I was very sceptical about the answer they gave in the series and researched a bit more. The answer in the show was correct, but the problem was not correctly explained.
The Monty Hall problem consists on, given 3 doors and only 1 containing a prize, picking the correct one. As it is random, you have 1/3 chances of winning, and 2/3 of losing. #probability #math
Follow me here on mastodon for posts on maths, physics, and especially #StochasticCalculus and #probability related material.
Also, check out my blog “Almost Sure”
I may or may not be posting on twitter (we’ll see how that goes…), but will cross-post any original content here.
Andrew Shtulman's “Reflecting on Possibility: Cognitive Reflection Facilitates the Development of Modal Cognition" replicated and extended finding that kinds think wrong actions are improbable, are improbable events are wrong!
Kids’ reflection test performance predicted kids possibility and permissibility judgments (above and beyond age and executive function).
Follow on gScholar to learn when article is up: https://scholar.google.com/citations?user=Cz8bTrkAAAAJ&hl=en&oi=ao
#cogSci #decisionScience #DevPsych #ethics #probability #xPhi
Kevin Dorst then presented “#Polarization is Not (Standard) #Bayesian”
Why not? Because most people violated Bayesian “Martingale updating”.
Why? “Intuitively, we often are (instrospectively) unsure what our probabilities/credences are."
Kevin proposes, “If we want adequate Bayesian models of polarization, they must incorporate introspection failures.”
See photo for the handout.
In the "Thinking about Social Groups" session of #SPP2023, David Kinney presented "Tell Me Your (Cognitive) Budget, and I’ll Tell You What You Value: ..." (in collaboration with Tania Lombrozo).
They tried to design stimuli that could test the degree to which people incorporate a representation of a problem, relevant data, and values.
I didn't see a preprint of this project online, but Dr. Kinney's on gScholar: https://scholar.google.com/citations?user=9gK2xIoAAAAJ&hl=en&oi=ao
@lulu_powerful so, now you'll have to define #likelihood for us, please?
@webmink In other words, when they designed the Eurostar they either didn’t imagine that it could get as #hot as it is, or they decided that designing to cope with it was too expensive, because the #probability of it happening was too low.
It’s probably the latter. Designing to be able to handle what was at the time maybe a 100 year event (or higher) is expensive. And the project was more concerned about the #Channel tunnel construction than if it would get hotter in the future. Now, depending on what the problem is, the #cost of fixing the issue will range from very #expensive to hideously expensive.
I hope the temperature goes down enough so you can get on the train soon.
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Hier ist noch der Link zum Interview mit Anton #Zeilinger, dem letztjährigen #PhysikNobelpreisträger:
#TheEconomist:
#Babbage: Untangling quantum mechanics with #NobelLaureate Anton Zeilinger
https://i.sonnet.fm/ijBvxdjLDmYqvjPf7
#QuantumEntanglement
#QuantumMechanics
#TheEconomist
#Vienna
#NobelPrize
#Physics
#Teleportation
#RichardFeynman
#AntonZeilinger
#Zeilinger
#photons
#probability
#AlbertEinstein
#Entanglement
Here's a normal distribution and logistic curve (really a derivative of a normal) as Powerpoint shapes.
Feel free to use it yourself:
http://steveharoz.com/public/gaussian.pptx
Here's a normal distribution and logistic curve as Powerpoint shapes.
Feel free to use it yourself:
http://steveharoz.com/public/gaussian.pptx
#probability : appearance of reality or truth
- French: Probabilité
- German: die Wahrscheinlichkeit
- Italian: probabilità
- Portuguese: probabilidade
- Spanish: probabilidad
------------
Thank you so much for being a member of our community!
Next was an interesting talk by my former advisor Sandy Pentland on understanding tail events in social systems at the Stanford Digital Economy Lab. This is an excellent combination of #probability and #networks, highly recommend https://www.youtube.com/watch?v=m1kn5jlcqow (3/8)
It's hard to estimate the odds of an asteroid impact from just a few observations. Early reports may indicate a high risk. Follow-ups usually reveal that the true path of the asteroid will miss Earth. That's what happened with asteroid 2023DW. (2/n)
#risk #probability
Want to think like a CIA analyst?
This short primer shares "structured analytic techniques" (SATs) to overcome common reasoning mistakes (see images):
- key assumptions check
- analysis of competing hypotheses
- alternative futures analysis
- and more!
There's also an outline (see final image) of the timeline of SATs for an intelligence report (e.g., to ensure faulty assumptions are addressed sooner than later).
#CIA #intelligence #CriticalThinking #logic #probability #bias #decisionScience
How do people go about writing automated tests around statistical properties? Example: we have a statistical test that produces the correct answer 90% of the time, how do you write a test around that? It seems to me that in the end, this implies a test that will fail about 10% of the time. I have been stumped by this problem, any suggestions appreciated!
#probability #simulation #statistics #testing
Yes, we need to become comfortable with uncertainty in our increasingly complex environment — in fact that should already have be the case, especially for issues that are complex enough to be 'interesting.'
And even more than being comfortable with uncertainty, we need to get much better at thinking of things in terms of likelihood and probability — even getting clear on the difference between these two terms. [Are you?]
So becoming comfortable with Bayesian thinking will equip us to more effectively deal with both uncertainty, and reasoning, as we try to make sense of the flood of social media, consumer-marketing, politics, and multiple flavors of bullshit.
And by the way, it also helps with the simpler, more straightforward stuff like new science, technology, and data…
#uncertainty #probability #likelihood # Bayesian #reason #reasoning #ToolsForThinking #complexity #sensemaking
> .. as statistician #ChristineAnnetteFranklin writes, #Nightingale was also a talented mathematician. In her twenties, she taught math to young children. She was particularly fascinated with #statistics and #probability An intensely religious person, she once said that “to understand God’s thoughts we must study statistics, for these are the measure of His purpose.”
https://daily.jstor.org/florence-nightingale-data-visualization-visionary/
#FlorenceNightingale
Another math birb in the series is posted:
Blog link: https://wordpress.com/post/fractalkitty.com/7905
I managed not to say they are in a bay/bayes - oops!
Who isn't tickled that the collective noun is an improbability for puffins?
#mathbirds #birbs #birdwatching #improbability #probability #bayestheorem #puffins #tuftedpuffin #oregon
#books to read:
This one from Jayne's builds up the rules of #probability theory from formal logic, rather than set theory as I learned it at uni.
It is dense, slow, and meaty reading.
It addresses shortcomings in how modern #science is practiced, that have bothered me since I got my first glimpse behind the curtain at the outset of my #phd.
A real framework for inference when using induction w/ incomplete information + data.
I'll be adding to this post over the coming weeks.
Study: Mexican jumping beans use random walk strategy to find shade - Enlarge / Mexican jumping beans are actually seed pods from a shrub nat... - https://arstechnica.com/?p=1912792 #mexicanjumpingbeans #statisticalphysics #probability #randomwalks #science #physics
It's time for me to add to the #introduction tag.
I'm Pete, a lecturer in Leeds, UK. In addition to mathematics, I'm a huge gadget nerd and an avid runner. I also never quite grew up and remain a child at heart.
#maths #probability #gadgets #apple #books #tech #geek #leeds
I found out that there is a DnD template for LaTeX, and now I am seriously considering writing a basic intro to probability for DnD players.
#DnD #Probability #StuffIWillNeverTrulyHaveTheTimeToDo
Boom 💥: birthday paradox checked this morning.
I have 35 students so without surprise we had 2 collisions. But they were intrigued by the reason behind it, so it was a success.
#education #eduglow #math #maths #mathematics #teaching #probability
#GameDesigners, when it comes to Dice, do you prefer Bell Curves or Equal Distributions? Comment bellow your reasonings!
Fun fact: you only need 23 people for there to be over 50% likelihood that two of them share the same birthday #probability