 Nov 2022

wexler.free.fr wexler.free.fr

Amos Tversky's famous "The Hot Hand in Basketball: On the Misperception of Random Sequences".


stats.stackexchange.com stats.stackexchange.com

The random process has outcomes
Notation of a random process that has outcomes
The "universal set" aka "sample space" of all possible outcomes is sometimes denoted by \(U\), \(S\), or \(\Omega\): https://en.wikipedia.org/wiki/Sample_space
Probability theory & measure theory
From what I recall, the notation, \(\Omega\), was mainly used in higherlevel grad courses on probability theory. ie, when trying to frame things in probability theory as a special case of measure theory things/ideas/processes. eg, a probability space, \((\cal{F}, \Omega, P)\) where \(\cal{F}\) is a \(\sigma\text{field}\) aka \(\sigma\text{algebra}\) and \(P\) is a probability density function on any element of \(\cal{F}\) and \(P(\Omega)=1.\)
Somehow, the definition of a sigmafield captures the notion of what we want out of something that's measurable, but it's unclear to me why so let's see where writing through this takes me.
Working through why a sigmaalgebra yields a coherent notion of measureable
A sigmaalgebra \(\cal{F}\) on a set \(\Omega\) is defined somewhat close to the definition of a topology \(\tau\) on some space \(X\). They're both collections of subcollections of the set/space of reference (ie, \(\tau \sub 2^X\) and \(\cal{F} \sub 2^\Omega\)). Also, they're both defined to contain their underlying set/space (ie, \(X \in \tau\) and \(\Omega \in \cal{F}\)).
Additionally, they both contain the empty set but for (maybe) different reasons, definitionally. For a topology, it's simply defined to contain both the whole space and the empty set (ie, \(X \in \tau\) and \(\empty \in \tau\)). In a sigmaalgebra's case, it's defined to be closed under complements, so since \(\Omega \in \cal{F}\) the complement must also be in \(\cal{F}\)... but the complement of the universal set \(\Omega\) is the empty set, so \(\empty \in \cal{F}\).
I think this might be where the similarity ends, since a topology need not be closed under complements (but probably has a special property when it is, although I'm not sure what; oh wait, the complement of open is closed in topology, so it'd be clopen! Not sure what this would really entail though 🤷♀️). Moreover, a topology is closed under arbitrary unions (which includes uncountable), but a sigmaalgebra is closed under countable unions. Hmm... Maybe this restriction to countable unions is what gives a coherent notion of being measurable? I suspect it also has to do with BanachTarski paradox. ie, cutting a sphere into 5 pieces and rearranging in a clever way so that you get 2 sphere's that each have the volume of the original sphere; I mean, WTF, if 1 sphere's volume equals the volume of 2 sphere's, then we're definitely not able to measure stuff any more.
And now I'm starting to vaguely recall that this what sigmafields essentially outlaw/ban from being possible. It's also related to something important in measure theory called a Lebeque measure, although I'm not really sure what that is (something about doing a Riemann integral but picking the partition on the yaxis/codomain instead of on the xaxis/domain, maybe?)
And with that, I think I've got some intuition about how fundamental sigmaalgebras are to letting us handle probability and uncertainty.
Back to probability theory
So then events like \(E_1\) and \(E_2\) that are elements of the set of subcollections, \(\cal{F}\), of the possibility space \(\Omega\). Like, maybe \(\Omega\) is the set of all possible outcomes of rolling 2 dice, but \(E_1\) could be a simple event (ie, just one outcome like rolling a 2) while \(E_2\) could be a compound(?) event (ie, more than one, like rolling an even number). Notably, \(E_1\) & \(E_2\) are NOT elements of the sample space \(\Omega\); they're elements of the powerset of our possibility space (ie, the set of all possible subsets of \(\Omega\) denoted by \(2^\Omega\)). So maybe this explains why the "closed under complements" is needed; if you roll a 2, you should also be able to NOT roll a 2. And the property that a sigmaalgebra must "contain the whole space" might be what's needed to give rise to a notion of a complete measure (conjecture about complete measures: everything in the measurable space can be assigned a value where that part of the measurable space does, in fact, represent some constitutive part of the whole).
But what about these "random events"?
Ah, so that's where random variables come into play (and probably why in probability theory they prefer to use \(\Omega\) for the sample space instead of \(X\) like a base space in topology). There's a function, that is, a mapping from outcomes of this "random event" (eg, a role of 2 dice) to a space in which we can associate (ie, assign) a sense of distance (ie, our sigmaalgebra). What confuses me is that we see things like "\(P(X=x)\)" which we interpret as "probability that our random variable, \(X\), ends up being some particular outcome \(x\)." But it's also said that \(X\) is a realvalued function, ie, takes some arbitrary elements (eg, events like rolling an even number) and assigns them a real number (ie, some \(x \in \mathbb{R}\)).
Aha! I think I recall the missing link: the notation "\(X=x\)" is really a shorthand for "\(X(\omega)=x\)" where \(\omega \in \cal{F}\). But something that still feels unreconciled is that our probability metric, \(P\), is just taking some real value to another real value... So which one is our sigmaalgebra, the inputs of \(P\) or the inputs of \(X\)? 🤔 Hmm... Well, I guess it has the be the set of elements that \(X\) is mapping into \(\mathbb{R}\) since \(X\text{'s}\) input is a small omega \(\omega\) (which is probably an element of big omega \(\Omega\) based on the conventions of small notation being elements of big notation), so \(X\text{'s}\) domain much be the sigmaalgrebra?
Let's try to generate a plausible example of this in action... Maybe something with an inequality like "\(X\ge 1\)". Okay, yeah, how about \(X\) is a random variable for the random process of how long it takes a customer to get through a grocery line. So \(X\) is mapping the elements of our sigmaalgebra (ie, what customers actually end up experiencing in the real world) into a subset of the reals, namely \([0,\infty)\) because their time in line could be 0 minutes or infinite minutes (geesh, 😬 what a life that would be, huh?). Okay, so then I can ask a question like "What's the probability that \(X\) takes on a value greater than or equal to 1 minute?" which I think translates to "\(P\left(X(\omega)\ge 1\right)\)" which is really attempting to model this whole "random event" of "What's gonna happen to a particular person on average?"
So this makes me wonder... Is this fact that \(X\) can model this "random event" (at all) what people mean when they say something is a stochastic model? That there's a probability distribution it generates which affords us some way of dealing with navigating the uncertainty of the "random event"? If so, then sigmaalgebras seem to serve as a kind of gateway and/or foundation into specific cognitive practices (ie, learning to think & reason probabilistically) that affords us a way out of being overwhelmed by our anxiety or fear and can help us reclaim some agency and autonomy in situations with uncertainty.

 Sep 2022

Local file Local file

B. V. Gnf.di.xko
Boris Vladimirovich Gnedenko https://en.wikipedia.org/wiki/Boris_Vladimirovich_Gnedenko
He wrote with/worked with Khinchin and Kolmogorov.

A . Y A . K H I N C H I N
Aleksandr Yakovlevich Khinchin https://en.wikipedia.org/wiki/Aleksandr_Khinchin


www.scientificamerican.com www.scientificamerican.com

Running this simulation over many time steps, Lilian Weng of OSoMe found that as agents' attention became increasingly limited, the propagation of memes came to reflect the powerlaw distribution of actual social media: the probability that a meme would be shared a given number of times was roughly an inverse power of that number. For example, the likelihood of a meme being shared three times was approximately nine times less than that of its being shared once.

 Aug 2022

eprint.iacr.org eprint.iacr.org1007.pdf1

Generating randomness.
EVM execution is deterministic. How to account for randomness?
Pesudo random generator, probability distribution.

 May 2022

docdrop.org docdrop.org

In the case ofLéviStrauss, meanwhile, the card index continued to serve inimportant ways as a ‘memory crutch’, albeit with a key differencefrom previous uses of the index as an aidememoire. In LéviStrauss’case, what the fallibility of memory takes away, the card index givesback via the workings of chance. As he explains in an interview withDidier Erebon:I get by when I work by accumulating notes – a bitabout everything, ideas captured on the fly,summaries of what I have read, references,quotations... And when I want to start a project, Ipull a packet of notes out of their pigeonhole anddeal them out like a deck of cards. This kind ofoperation, where chance plays a role, helps merevive my failing memory. (Cited in Krapp, 2006:361)For Krapp, the crucial point here is that, through his use of indexcards, LéviStrauss ‘seems to allow that the notes may either restorememory – or else restore the possibilities of contingency which givesthinking a chance under the conditions of modernity’ (2006: 361).
Claude LéviStrauss had a note taking practice in which he accumulated notes of ideas on the fly, summaries of what he read, references, and quotations. He kept them on cards which he would keep in a pigeonhole. When planning a project, he would pull them out and use them to "revive [his] failing memory."
Questions:  Did his system have any internal linkages?  How big was his system? (Manageable, unmanageable?)  Was it only used for memory, or was it also used for creativity?  Did the combinatorial reshufflings of his cards provide inspiration a la the Llullan arts?
Link this to the ideas of Raymond Llull's combinatorial arts.

 Apr 2022

twitter.com twitter.com

ReconfigBehSci. (2021, February 1). @islaut1 @richarddmorey I think of strength of inference resting on P(not Enot H) (for coronavirus case). Search determines the conditional probability (and by total probability of course prob of evidence) but it isn’t itself the evidence. So, was siding with R. against what I thought you meant ;) [Tweet]. @SciBeh. https://twitter.com/SciBeh/status/1356216290847944706

 Jan 2022

bio.libretexts.org bio.libretexts.org

If the round pea parent is heterozygous, there is a oneeighth probability that a random sample of three progeny peas will all be round.
Please clarify how you calculated 1:8 ratio. One plant has rr and the other plant has Rr. You create a table and get two Rr and two rr which makes the probability of getting three Rr or RR zero.

 Dec 2021

www.medrxiv.org www.medrxiv.org

Lai, J., German, J., Hong, F., Tai, S.H. S., McPhaul, K. M., Milton, D. K., & Group, for the U. of M. S. R. (2021). Comparison of Saliva and MidTurbinate Swabs for Detection of COVID19 (p. 2021.12.01.21267147). https://doi.org/10.1101/2021.12.01.21267147

 Oct 2021

www.washingtonpost.com www.washingtonpost.com

Washington Post. ‘Opinion  Remaining Unvaccinated in Public Should Be Considered as Bad as Drunken Driving’, 15 September 2021. https://www.washingtonpost.com/opinions/2021/09/15/remainingunvaccinatedpublicshouldbeconsideredbaddrunkendriving/.

 Sep 2021

twitter.com twitter.com

Sam Wang on Twitter: “These are risk levels that you pose to other people. They’re compared with you as—A nonsmoker—A sober driver—A vaccinated person. Unvaccinated? 5x as likely to get sick, for 3x as long. Total risk to others? 15x a vaccinated person Details:https://t.co/ckTWaivK8n https://t.co/PhpLvX2dsm” / Twitter. (n.d.). Retrieved September 19, 2021, from https://twitter.com/SamWangPhD/status/1438361144759132167

 Aug 2021

twitter.com twitter.com

Tim Plante, MD MHS on Twitter: “Just reported: About half of recent ICU patients with #Covid19 in #Vermont are vaccinated. Sounds like the vaccines aren’t working, right? WRONG. Vaccines are working and here’s why. But first, let’s talk a bit about unprotected sex. A thread. (Refs at the end.) 1/n https://t.co/iyQcfCDAfh” / Twitter. (n.d.). Retrieved August 27, 2021, from https://twitter.com/tbplante/status/1430222978961317896


journals.sagepub.com journals.sagepub.com

Sun, Q., Lu, J., Zhang, H., & Liu, Y. (2021). Social Distance Reduces the Biases of Overweighting Small Probabilities and Underweighting Large Probabilities. Personality and Social Psychology Bulletin, 47(8), 1309–1324. https://doi.org/10.1177/0146167220969051

 Jul 2021

psyarxiv.com psyarxiv.com

Moore, D. A., Backus, M., & Little, A. T. (2021). Constraints on Thinking Cause Overprecision [Preprint]. PsyArXiv. https://doi.org/10.31234/osf.io/evcx2

 Jun 2021

science.sciencemag.org science.sciencemag.org

Cheng, Y., Ma, N., Witt, C., Rapp, S., Wild, P. S., Andreae, M. O., Pöschl, U., & Su, H. (2021). Face masks effectively limit the probability of SARSCoV2 transmission. Science. https://doi.org/10.1126/science.abg6296

 May 2021

80000hours.org 80000hours.org

think of your career as a series of experiments designed to help you learn about yourself and test out potentially great longerterm paths
I wonder if there's a connection here to Duke, A. (2019). Thinking in Bets: Making Smarter Decisions When You Don’t Have All the Facts. Portfolio.
I haven't read the book but it's on my list.


twitter.com twitter.com

Dr Ellie Murray. (2021, May 7). I’m seeing a lot of “these people are overestimating risk” chatter that doesn’t acknowledge that the probability you die if you get covid is always less than the probability anyone dies if you get covid. It’s not “overestimation” to consider community impacts. [Tweet]. @EpiEllie. https://twitter.com/EpiEllie/status/1390792624777334797

 Apr 2021

stats.libretexts.org stats.libretexts.org

Events AAA and BBB are mutually exclusive (cannot both occur at once) if they have no elements in common.
Events \(A\) and \(B\) are mutually exclusive (cannot both occur at once) if they have no elements in common.
Events \(A\) and \(B\) are mutually exclusive if: $$P(A∩B)=0$$

The complement of an event AAA in a sample space SSS, denoted AcAcA^c, is the collection of all outcomes in SSS that are not elements of the set AAA. It corresponds to negating any description in words of the event AAA.
The complement of an event \(A\) in a sample space \(S\), denoted \(A^c\), is the collection of all outcomes in \(S\) that are not elements of the set \(A\). It corresponds to negating any description in words of the event \(A\).
The complement of an event \(A\) consists of all outcomes of the experiment that do not result in event \(A\).
Complement formula:
$$P(A^c)=1P(A)$$
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seeingtheory.brown.edu seeingtheory.brown.edu
 Mar 2021

www.namics.nysaes.cornell.edu www.namics.nysaes.cornell.edu

In each of the games moves are entirely determined by chance; there is no opportunity to make decisions regarding play. (This, of course, is one reason why most adults with any intellectual capacity have little interest in playing the games for extended times, especially since no money or alcohol is involved.)
The entire motivation for this study.



Devriendt, K., MartinGutierrez, S., & Lambiotte, R. (2020). Variance and covariance of distributions on graphs. ArXiv:2008.09155 [Physics, Stat]. http://arxiv.org/abs/2008.09155


onlinelibrary.wiley.com onlinelibrary.wiley.com

Traczyk, J., Fulawka, K., Lenda, D., & Zaleskiewicz, T. (n.d.). Consistency in probability processing as a function of affective context and numeracy. Journal of Behavioral Decision Making, n/a(n/a). https://doi.org/10.1002/bdm.2206



<small><cite class='hcite via'>ᔥ <span class='pauthor hcard'>hyperlink.academy</span> in The Future of Textbooks (<time class='dtpublished'>03/18/2021 23:54:19</time>)</cite></small>


arxiv.org arxiv.org

Hota, Ashish R., Tanya Sneh, and Kavish Gupta. ‘Impacts of GameTheoretic Activation on Epidemic Spread over Dynamical Networks’. ArXiv:2011.00445 [Physics], 1 November 2020. http://arxiv.org/abs/2011.00445.


danallosso.substack.com danallosso.substack.com

He introduces the idea of the apophatic: what we can't put into words, but is important and vaguely understood. This term comes from Orthodox theology, where people defined god by saying what it was not.
Too often as humans we're focused on what is immediately in front of us and not what is missing.
This same thing plagues our science in that we're only publishing positive results and not negative results.
From an information theoretic perspective, we're throwing away half (or more?) of the information we're generating. We might be able to go much farther much faster if we were keeping and publishing all of our results in better fashion.
Is there a better word for this negative information? #openquestions


kids.frontiersin.org kids.frontiersin.org


Unrealistic optimism about future life events: A cautionary note. (n.d.). Retrieved March 4, 2021, from https://psycnet.apa.org/fulltext/201022979001.pdf?auth_token=a25fd4b7f008a50b15fd7b0f1fdb222fc38373f4

 Feb 2021

academic.oup.com academic.oup.com

Westreich, D., & Iliinsky, N. (2014). Epidemiology Visualized: The Prosecutor’s Fallacy. American Journal of Epidemiology, 179(9), 1125–1127. https://doi.org/10.1093/aje/kwu025



Hanea, A., Wilkinson, D. P., McBride, M., Lyon, A., Ravenzwaaij, D. van, Thorn, F. S., Gray, C. T., Mandel, D. R., Willcox, A., Gould, E., Smith, E., Mody, F., Bush, M., Fidler, F., Fraser, H., & Wintle, B. (2021). Mathematically aggregating experts’ predictions of possible futures. MetaArXiv. https://doi.org/10.31222/osf.io/rxmh7

 Dec 2020

cims.nyu.edu cims.nyu.edu

gromov
Mikhael Gromov's docs at NYU
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chem.libretexts.org chem.libretexts.org

Nodes A wave function node occurs at points where the wave function is zero and changes signs. The electron has zero probability of being located at a node.
Nodes
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 Oct 2020

seeingtheory.brown.edu seeingtheory.brown.edu

Kunin, D. (n.d.). Seeing Theory. Retrieved October 27, 2020, from http://seeingtheory.io


www.youtube.com www.youtube.com

David Spiegelhalter and False Positives. (2020, October 14). https://www.youtube.com/watch?v=XmiEzi54lBI&feature=youtu.be


www.agweb.com www.agweb.com

the hand of God
read "statistical mechanics"


parenting.nytimes.com parenting.nytimes.com

How this phenomenon translates into absolute, rather than relative, risk, however, is a bit thorny. A large study published in 2018, for instance, found that among women who had children between 34 and 47, 2.2 percent developed breast cancer within three to seven years after they gave birth (among women who never had children, the rate was 1.9 percent). Over all, according to the American Cancer Society, women between 40 and 49 have a 1.5 percent chance of developing breast cancer.
The rates here are so low as to be nearly negligible on their face. Why bother reporting it?


www.dpmms.cam.ac.uk www.dpmms.cam.ac.uk

This result of Erd ̋os [E] is famous not because it has large numbers of applications,nor because it is difficult, nor because it solved a longstanding open problem. Its famerests on the fact that it opened the floodgates to probabilistic arguments in combinatorics.If you understand Erd ̋os’s simple argument (or one of many other similar arguments) then,lodged in your mind will be a general principle along the following lines:if one is trying to maximize the size of some structure under certain constraints, andif the constraints seem to force the extremal examples to be spread about in a uniformsort of way, then choosing an example randomly is likely to give a good answer.Once you become aware of this principle, your mathematical power immediately increases.


twitter.com twitter.com

Nick Brown on Twitter. (n.d.). Twitter. Retrieved October 1, 2020, from https://twitter.com/sTeamTraen/status/1311282470084644865

 Sep 2020

advances.sciencemag.org advances.sciencemag.org

Holman, E. A., Thompson, R. R., Garfin, D. R., & Silver, R. C. (2020). The unfolding COVID19 pandemic: A probabilitybased, nationally representative study of mental health in the U.S. Science Advances, eabd5390. https://doi.org/10.1126/sciadv.abd5390

 Aug 2020

www.medrxiv.org www.medrxiv.org

Ray, E. L., Wattanachit, N., Niemi, J., Kanji, A. H., House, K., Cramer, E. Y., Bracher, J., Zheng, A., Yamana, T. K., Xiong, X., Woody, S., Wang, Y., Wang, L., Walraven, R. L., Tomar, V., Sherratt, K., Sheldon, D., Reiner, R. C., Prakash, B. A., … Consortium, C.19 F. H. (2020). Ensemble Forecasts of Coronavirus Disease 2019 (COVID19) in the U.S. MedRxiv, 2020.08.19.20177493. https://doi.org/10.1101/2020.08.19.20177493


covid19.iza.org covid19.iza.org

Explaining Governors’ Response to the COVID19 Pandemic in the United States. COVID19 and the Labor Market. (n.d.). IZA – Institute of Labor Economics. Retrieved August 8, 2020, from https://covid19.iza.org/publications/dp13137/


www.nber.org www.nber.org

Lo, A. W., Siah, K. W., & Wong, C. H. (2020). Estimating Probabilities of Success of Vaccine and Other AntiInfective Therapeutic Development Programs (Working Paper No. 27176; Working Paper Series). National Bureau of Economic Research. https://doi.org/10.3386/w27176


www.bbc.co.uk www.bbc.co.uk

BBC Radio 4—The Political School, Episode 1. (n.d.). BBC. Retrieved August 2, 2020, from https://www.bbc.co.uk/programmes/m000kv6v

 Jul 2020

osf.io osf.io

Andia, T., Mantilla, C., RodriguezLesmes, P., Criado, L., Gomez, J. S., Ortiz, S., Quintero, A., Rincón, H., & Romero, S. (2020). Mentioning anosmia improves how community pharmacies handle phone call requests during the COVID19 pandemic: An audit study in Colombia [Preprint]. SocArXiv. https://doi.org/10.31235/osf.io/s2z47


www.nature.com www.nature.com

Block, P., Hoffman, M., Raabe, I. J., Dowd, J. B., Rahal, C., Kashyap, R., & Mills, M. C. (2020). Social networkbased distancing strategies to flatten the COVID19 curve in a postlockdown world. Nature Human Behaviour, 4(6), 588–596. https://doi.org/10.1038/s4156202008986


www.sciencedirect.com www.sciencedirect.com

Argument Quality in Real World Argumentation. (2020). Trends in Cognitive Sciences, 24(5), 363–374. https://doi.org/10.1016/j.tics.2020.01.004

 Jun 2020

twitter.com twitter.com

ReconfigBehSci on Twitter: “Fellow behavioural scientists, I had a lightbulb moment yesterday. I suspect I might have been unusually slow here, and you all clocked this months ago, but thought I’d share nevertheless: I’ve lost count of how often I’ve been told ‘You only have a 1/100 chance of dying’” / Twitter. (n.d.). Twitter. Retrieved June 27, 2020, from https://twitter.com/scibeh/status/1276434777856446464


psyarxiv.com psyarxiv.com

Veltri, G. A., Prof, LupiáñezVillanueva, F., Folkvord, F., Theben, A., & Gaskell, G. (2020, April 29). The impact of online platform transparency of information on consumer’s choices. https://doi.org/10.31234/osf.io/htja5


psycnet.apa.org psycnet.apa.org

Attali, Y., Budescu, D., & ArieliAttali, M. (2020). An item response approach to calibration of confidence judgments. Decision, 7(1), 1–19. https://doi.org/10.1037/dec0000111


psycnet.apa.org psycnet.apa.org

Winman, A., Hansson, P., & Juslin, P. (2004). Subjective Probability Intervals: How to Reduce Overconfidence by Interval Evaluation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 30(6), 1167–1175. https://doi.org/10.1037/02787393.30.6.1167


psycnet.apa.org psycnet.apa.org

Erev, I., Wallsten, T. S., & Budescu, D. V. (1994). Simultaneous over and underconfidence: The role of error in judgment processes. Psychological Review, 101(3), 519–527. https://doi.org/10.1037/0033295X.101.3.519

 May 2020

www.sjdm.org www.sjdm.org

[Jdmsociety] Decision on reopening economies. (n.d.). Retrieved April 21, 2020, from http://www.sjdm.org/mailarchive/jdmsociety/2020April/008496.html


psyarxiv.com psyarxiv.com

Dhami, M. K., & Mandel, D. R. (2020). UK and US policies for communicating probability in intelligence analysis: A review [Preprint]. PsyArXiv. https://doi.org/10.31234/osf.io/kuyhb


psyarxiv.com psyarxiv.com

Moore, D. A., & Wallsten, T. S. (2020). Rolling Forecasts. https://doi.org/10.31234/osf.io/ryvg3


twitter.com twitter.com

Dr Muge Cevik on Twitter
Tags
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 household
 attack rate
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 is:twitter
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threadreaderapp.com threadreaderapp.com

Thread by @taaltree: Antibody tests are coming online. Never before have humans needed to understand Bayes rule more. Let’s talk about why it’s critical NOT to a…. (n.d.). Retrieved April 21, 2020, from https://threadreaderapp.com/thread/1248467731545911296.html

 Apr 2020

psyarxiv.com psyarxiv.com

Mandel, D. R., Wallsten, T. S., & Budescu, D. (2020, April 20). NumericallyBounded Language Schemes Are Unlikely to Communicate Uncertainty Effectively. https://doi.org/10.31234/osf.io/9f6ev


math.stackexchange.com math.stackexchange.com

Therefore, En=2n+1−2=2(2n−1)
Simplified formula for the expected number of tosses (e) to get
n
consecutive heads(n≥1)
:$$e_n=2(2^n1)$$
For example, to get 5 consecutive heads, we've to toss the coin 62 times:
$$e_n=2(2^51)=62$$
We can also start with the longer analysis of the 5 scenarios:
 If we get a tail immediately (probability 1/2) then the expected number is e+1.
 If we get a head then a tail (probability 1/4), then the expected number is e+2.
 If we get two head then a tail (probability 1/8), then the expected number is e+2.
 If we get three head then a tail (probability 1/16), then the expected number is e+4.
 If we get four heads then a tail (probability 1/32), then the expected number is e+5.
 Finally, if our first 5 tosses are heads, then the expected number is 5.
Thus:
$$e=\frac{1}{2}(e+1)+\frac{1}{4}(e+2)+\frac{1}{8}(e+3)+\frac{1}{16}\\(e+4)+\frac{1}{32}(e+5)+\frac{1}{32}(5)=62$$
We can also generalise the formula to:
$$e_n=\frac{1}{2}(e_n+1)+\frac{1}{4}(e_n+2)+\frac{1}{8}(e_n+3)+\frac{1}{16}\\(e_n+4)+\cdots +\frac{1}{2^n}(e_n+n)+\frac{1}{2^n}(n) $$


www.cmu.edu www.cmu.edu

Fischhoff, B., de Bruin, W. B., Güvenç, Ü., Caruso, D., & Brilliant, L. (2006). Analyzing disaster risks and plans: An avian flu example. Journal of Risk and Uncertainty, 33(1–2), 131–149. https://doi.org/10.1007/s1116600601758

 Jan 2020

levels.io levels.io

My friend Marc again to the rescue. He suggested that since there was 10,000+ people RT'ing and following, I could just pick a random follower from my current total follower list (78,000 at this point), then go to their profile to check if they RT'd it and see. If they didn't, get another random follower and repeat, until you find someone. With 78,000 followers this should take about ~8 tries.
Technically he said it would be random among those who retweeted, but he's chose a much smaller subset of people who are BOTH following him and who retweeted it. Oops!
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 Dec 2019

en.wikipedia.org en.wikipedia.org

the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.


colinwalker.blog colinwalker.blog

Many people luck out like me, accidentally. We recognize what particular path to mastery we’re on, long after we actually get on it.
Far too many people luck out this way and we all perceive them as magically talented when in reality, they're no better than we, they just had better circumstances or were in the right place at the right time.

 Mar 2019

complexityzoo.uwaterloo.ca complexityzoo.uwaterloo.ca

Special Complexity Zoo Exhibit: Classes of Quantum States and Probability Distributions 24 classes and counting! A whole new phylum of the Complexity kingdom has recently been identified. This phylum consists of classes, not of problems or languages, but of quantum states and probability distributions. Well, actually, infinite families of states and distributions, one for each number of bits n. Admittedly, computer scientists have been talking about the complexity of sampling from probability distributions for years, but they haven't tended to organize those distributions into classes designated by inscrutable sequences of capital letters. This needs to change.

 Feb 2019

paleorxiv.org paleorxiv.org

n they will share similar genes, but it 18is the phenotype –upon which selection acts –which is crucia
There two important things to note.
If the same genetic programme leads to two phenotypes because of the environment, this falls in the category of epigenetics. Epigenetic processes are usually not treelike, hence, poorly modelled by inferring a tree.
You implicitly assume (via your Rscript) that homoiologies (in a strict sense, i.e. parallelism) are rare and not beneficial (neutral). But if the homoiology is beneficial (i.e. positively selected for), it will be much more common in a clade of close relatives than the primitive phenotype (the symplesiomorphy). We can further assume that beneficial homoiologies will accumulate in the mostderived, advanced, specialised taxa, in the worst case (from the mainstream cladistic viewpoint) mimicking or even outcompeting synapomorphies. A simply thought example: let's say we have a monophylum (fide Hennig) with two sublineages, each sublineage defined by a single synapormorphy. Both sublineages radiate and invade in parallel a new niche (geographically separated from each other) and fix (evolve) a set of homoiologies in adaptation to that new niche. The members of both sublineages with the homoiologies will be resolved as one clade, a pseudomonophylum, supported by the homoiologies as pseudosynapomorphies. And the actual synapomorphies will be resolved as plesiomorphies or autapomorphies.
Without molecular (and sometime even with, many molecular trees are based on plastid in plants and mitochondria in animals, and both are maternally inherited, hence, geographically controlled) or ontologicalphysiological control it will be impossible to make a call what is derived (hence a potential homoiology) and what ancestral in a group of organisms sharing a relative recent common origin and a still similiar genetic programme.

 Oct 2018

chem.libretexts.org chem.libretexts.org

In contrast to his concept of a simple circular orbit with a fixed radius, orbitals are mathematically derived regions of space with different probabilities of having an electron.
In this case, the QM model allows for probabilistic radii, not fixed radii, and the quantization is the energy level. An electron with principal quantum number n = 2 will always have quantized energy corresponding to \( E = R(1/n^2) \), but the exact minimal and maximal radial distance from the nucleus is not specified as in the Bohr model of the atom. Similar to the Bohr model though, the most probable radial distance is quantifiable, and that is the radius the electron is most likely to inhabit, however it will be found elsewhere at other times.

 Sep 2017

thebulletin.org thebulletin.org

Terrorist use of an actual nuclear bomb is a lowprobability event
Low probability and high impact but not a black swan

 Feb 2017

static1.squarespace.com static1.squarespace.com

These two qualities, therefore, PROBABILITY and PLAUSIBILITY
This is an important set of terms to think through in terms of come to think about and with rhetoric.

CHAPTER VI
Chapter VII: General Audience Awareness
But, really, Mere Rhetoric has a nice (I'm assuming she's mostly on point here) summary of some of the concepts to follow.

 Jan 2017

static1.squarespace.com static1.squarespace.com

Hume considers the possibility that there is, indeed, complete relativism in this matter. But his purpose is to find ways to reduce or eliminate disagreement, to set a standard
A rhetorical concern dating back to at least Aristotle: how to decide upon things in the realm of the probable rather than the absolute.

 Nov 2016

journals.plos.org journals.plos.org

Finally, by assuming the nondetection of a species to indicate absence from a given grid cell, we introduced an extra level of error into our models. This error depends on the probability of false absence given imperfect detection (i.e., the probability that a species was present but remained undetected in a given grid cell [73]): the higher this probability, the higher the risk of incorrectly quantifying speciesclimate relationships [73].
This will be an ongoing challenge for species distribution modeling, because most of the data appropriate for these purposes is not collected in such a way as to allow the straightforward application of standard detection probability/occupancy models. This could potentially be addressed by developing models for detection probability based on species and habitat type. These models could be built on smaller/different datasets that include the required data for estimating detectability.

 Jul 2016

www.quantamagazine.org www.quantamagazine.org

hyperuniform distribution  Appears random at smaller scales, but more predictable at larger scales.

 Feb 2016

blog.cloudera.com blog.cloudera.com

Great explanation of 15 common probability distributions: Bernouli, Uniform, Binomial, Geometric, Negative Binomial, Exponential, Weibull, Hypergeometric, Poisson, Normal, Log Normal, Student's t, ChiSquared, Gamma, Beta.

 Jan 2016

blogs.scientificamerican.com blogs.scientificamerican.com

P(BE) = P(B) X P(EB) / P(E), with P standing for probability, B for belief and E for evidence. P(B) is the probability that B is true, and P(E) is the probability that E is true. P(BE) means the probability of B if E is true, and P(EB) is the probability of E if B is true.


phys.org phys.org

paradox of unanimity  Unanimous or nearly unanimous agreement doesn't always indicate the correct answer. If agreement is unlikely, it indicates a problem with the system.
Witnesses who only saw a suspect for a moment are not likely to be able to pick them out of a lineup accurately. If several witnesses all pick the same suspect, you should be suspicious that bias is at work. Perhaps these witnesses were cherrypicked, or they were somehow encouraged to choose a particular suspect.

 Oct 2015


Nearly all applications of probability to cryptography depend on the factor principle (or Bayes’ Theorem).
This is easily the most interesting sentence in the paper: Turing used Bayesian analysis for codebreaking during WWII.

 Oct 2013

rhetoric.eserver.org rhetoric.eserver.org

Now the propositions of Rhetoric are Complete Proofs, Probabilities, and Signs.
From chapter 2.


rhetoric.eserver.org rhetoric.eserver.org

the materials of enthymemes are Probabilities and Signs


rhetoric.eserver.org rhetoric.eserver.org

The premisses from which enthymemes are formed are "probabilities" and "signs"
