Introduction to probability there are many resources in videos that tell you how to calculate a value. For a probability.
This video doesn’t do that.
This video explains what probability is, how probability is used and ways that probability can be estimated.
Much that happens in our world is not completely predictable, but we often estimate How likely an unpredictable event is.
We draw on previous experience to estimate How likely something in the future is to occur.
We express the likelihood or probability is a number between zero and one where an event with a probability of zero will not happen and an event with a probability of one must happen.
An event of probability 5 is just as likely to happen as not to happen when we use words like very likely a fair chance or might happen.
The message can get confused.
Using numbers makes things clearer, even though our perceptions might be different probability, values are used to communicate.
How likely something is probabilities are also used in mathematical models. These models are used for many things, such as predicting weather, including floods and tornadoes deciding how many seats to sell at a certain certain price on a flight, creating computer games and evaluating forensic evidence.
Probability, values are just estimates of How likely it is for something to happen.
They do not tell us whether they will happen or not.
When we can list all the equally likely outcomes, it can be easy to estimate the probability.
These are sometimes called exact probabilities a priori traditional or model estimates other times s.
We use historical data or run an experiment to estimate the probability.
We assume that the future Behavior will be similar to previous Behavior.
We need to have many instances from the past or an experiment in order to have a good estimate of the probability.
This source of probability is called relative frequency, historical or experimental.
Sometimes we do not have enough information, and so we assign a probability value according to how we think things will turn out. These can be quite unreliable, but unfortunately, can be given a lot of influence because they seem official to find out more about probability watch our other videos.
This video was brought to you by statistics.
Learning Center visit our website for more resources to help you learn and subscribe to this channel to find out .
Thanks to CuriosityStream for sponsoring this video, Everyone has an intuitive understanding of what the viscosity of a fluid is. Honey is more viscous than oil, and oil is more viscous than water. Viscous fluids feel thicker and don’t flow as easily, But since it’s such a fundamental parameter that defines how fluids behave, it’s worth developing a more in depth. Understanding of viscosity To do this. Let’s start by looking at fluid flowing over a flat surface. It’s sometimes useful to think of fluid, as flowing in layers with each layer moving at a different velocity When two layers are moving relative to one another, because they’re flowing at different velocities, a shear stress, develops between them. This is similar to how a frictional force develops between two solid objects sliding relative to one another. The effect of this shear stress is most obvious in flow close to a wall. The wall imparts a large shear stress onto the fluid particles that are in contact with it, causing them to have zero velocity. This is called the no sIip condition. The large shear stress is transmitted through to adjacent layers of flow, slowing down flow close to the wall and resulting in a velocity profile. That looks like this. The magnitude of the shear stress, acting between the layers of the fluid, is closely linked to the slope of the velocity profile du dy. Where? U is the fluid velocity and y, is the distance from the wall? The slope is small in the free stream, The fluid layers are moving at almost exactly the same velocity, and so the shear stress will also be very small But close to the wall. The velocity changes very suddenly In this region. The slope is large, and so the shear stress is also large For most fluids. The relationship between the shear stress and the slope of the velocity profile is linear, And the constant of proportionality is what we call fluid viscosity In engineering. It’s usually denoted using the Greek letter Mu. We can think of viscosity as the internal friction of a fluid in motion. It has the effect of smoothing out differences in velocity by increasing the shear stresses in proportion with the velocity gradient. We can express the linear relationship using this equation. This is called Newton’s, Law of Viscosity and fluids that obey it are said to be Newtonian. It describes how easily a fluid will flow. The viscosity Mu is sometimes called the dynamic viscosity, as this differentiates it from the kinematic viscosity, which is the dynamic viscosity divided by the fluid density. Let’s look at how a small fluid element like this one will deform as it flows. If du dy is positive, the upper surface will move faster than the lower surface and so, over a time period, Delta T the upper surface will travel further by a distance equal to Delta. U times Delta T This generates a shear strain that we can calculate using trigonometry and the small angle approximation By re arranging. We can see that the slope of the velocity profile is equivalent to the rate at which the shear strain is applied. So now we can write Newton’s law of Viscosity in a different form like this. This looks a lot like Hooke’s law for shear that we see in solids, except instead of being a function of the strain. The shear stress is a function of the rate at which the strain is applied. This makes sense if we consider how solids and fluids respond to applied. Shear forces Solids respond to a constantly applied shear force with a finite amount of deformation, whereas fluids respond by deforming continuously, they flow for as long as the shear force is applied. This is why, for fluids, the shear stress is a function of the rate at which strain changes over time, instead of being a function of the shear strain itself like it is for solids. If we do some quick dimensional analysis on Newton’s law of viscosity, we can see that the dynamic viscosity has units of Pascal seconds And the kinematic viscosity has units of meters squared per second. The Poise and Stokes are other units for the dynamic and kinematic viscosities, although they are often expressed as centipoise and centistokes. The centipoise unit is particularly convenient because at room temperature, the viscosity of water is equal to one centipoise Typical engine oil at room temperature has a viscosity of around 500 centipoise, so is 500 times more viscous than water. The viscosity of honey is ten thousand centipoise, And if we keep going, we get to fluids that are so viscous they appear to be solid Pitch is a thick tar like fluid that has extremely high viscosity, as is famously demonstrated by the pitch drop experiment In 1927. At the University of Queensland, heated pitch was added to a funnel and once it cooled it was allowed to drop into a beaker placed below it Because of its extremely high viscosity. Only nine drops have fallen into the beaker since then. It is the world’s longest, running continuous, lab experiment. On the other end of the spectrum we have gases which generally have much lower viscosity than liquids. Air has a viscosity of 0 018 centipoise at room temperature, for example. One very interesting substance is Helium 4, the common isotope of helium, A gas under normal conditions. It becomes a liquid when cooled to below 4 Kelvin and at these low temperatures researchers have found that it can behave like it has zero viscosity Substances that behave in this way are called superfluids, So we know how to define viscosity, but what actually causes it on The molecular level In liquids viscosity is caused by inter molecular cohesive forces which bond layers of the fluid together And in gases it’s caused by inter molecular collisions, which create interaction between adjacent layers of the fluid Viscosity is highly dependent on temperature. In liquids. It decreases as temperature increases. This is because, at the molecular level, an increased temperature allows molecules to more easily escape the attractive forces of the adjacent molecules. The viscosity of water, for example, reduces from 1 centipoise at 20 degrees Celsius to 0 5 centipoise at 50 degrees Celsius, But temperature has the opposite effect in gases. Viscosity increases with increasing temperature, because the higher temperature means that gas molecules have more random motion which results in more inter molecular collisions. The temperature dependence of viscosity can be modelled using simple empirical correlations. We can use Andrade’s equation for liquids and Sutherland’s equation for gases, where the various constants are determined for each fluid by experimentation. Viscosity also varies with pressure, although to a much lesser extent than temperature and so pressure dependence is often neglected Because it fundamentally affects how fluids behave. Viscosity is a very important parameter in Fluid Mechanics. It appears in the equation for Reynolds number, for example, and so affects whether flow is likely to be laminar or turbulent Flow of high viscosity fluids is more likely to be laminar, because any small turbulent disturbances are more easily dampened out by the larger shear stresses Viscosity. Also causes the pressure drop along a pipe Without viscosity. There would be no shear stresses imparted by the pipe wall, and the velocity profile would look like this. The existence of viscosity can make modelling flow quite complicated, and so engineers often try to neglect viscous forces where possible. The assumption that viscous forces are negligible is typically only applied to certain regions within a larger flow system. It’s never valid for flow close to a boundary, for example, where viscous forces are always significant, But outside of the boundary layer at high Reynolds numbers, viscous forces can often be neglected Flow where viscous forces can be neglected is said to be inviscid. Assuming that flow is inviscid, doesn’t mean we’re, assuming the fluid has no viscosity. It just means that we’re neglecting viscous forces because they’re small compared to other forces. Doing this makes analysis of the flow much easier. The Navier Stokes equations define the behaviour of fluids, but are very difficult to solve. Assuming that flow is inviscid means that we can neglect the viscous terms which contain the higher order. Derivatives, The resulting equations for inviscid flow are called the Euler equations and they’re much easier to solve. Inviscid flow is also a key assumption in the derivation of Bernoulli’s equation. We said earlier that fluids that obey Newton’s law of viscosity are said to be Newtonian. The shear stress increases linearly with the strain rate, But for some fluids the relationship between shear stress and strain rate is non linear. These are called non Newtonian fluids and there are two main types: shear thickening fluids and shear thinning fluids For shear thickening fluids. The apparent viscosity increases as the strain rate increases And for shear thinning fluids. The apparent viscosity reduces as the strain rate increases Paint, is an example of a shear thinning fluid It’s easily applied with a brush, but doesn’t drip from the wall once it has been applied. Viscosity really is a fundamental parameter in the study of fluids. Engine oil is one example of an application where viscosity and the effect of temperature play crucial roles. I’ve included a bit more discussion about engine oil viscosity in the extended version of this video over on Nebula Nebula is a streaming platform built by independent educational creators. It’s a place where you can find loads of amazing original content. Like Mustard’s, The Origins of Stealth that explores the history of the F 117 Nighthawk, But it also has extended versions of videos from your favourite creators, with what we’re calling Nebula. The Nebula version of this video includes a few extra minutes on engine oil viscosity, and that content replaces this sponsor segment, since Nebula is completely ad free To make Nebula even better. We’ve teamed up with CuriosityStream. Curiositystream is the best place to go for high quality documentaries. It has thousands of titles like Autonomy, a film that explores the implications of us, giving up control to self driving cars or the series A World Without NASA that examines how technologies developed by the space agency affect out everyday lives. So to get access to all of the great documentaries on CuriosityStream and my bonus content on Nebula head over to curiositystream com, efficientengineer or click on the link. In the description You’ll get a 26 discount on the CuriosityStream annual plan AND you’ll get Nebula for free All for less than 15 dollars a year. It’s an amazing deal and signing up is a great way to support this channel And that’s it for this look at fluid viscosity Thanks for watching
the last of us has always been a game that in some way or other is defined by its tech this is probably not the way its creators want you the player thinking about it they'd rather have you in the moment envisioning this rich vibrant narrative experience at the heart of a tense survival horror stealth action game and don't get me wrong it is that it's genuinely one of the best games ever made in my opinion but to me so much of its lasting success lies in those nuts and bolts as much as it does in the script or the mechanics a great deal of its narrative impact for example comes from those little moments those palpably pregnant pauses where a character won't say anything but an errant glance a broken watch will send all kinds of questions cascading through your head about what that character is thinking a glance whose power whose beautiful ambiguity is arguably only made possible thanks to naughty dog's industry-leading facial animation and capture tech rivaled only by a company like rockstar in the detail on show all this to say that when you have a game whose main pool is its storytelling both in cut scenes and its often frantic gameplay and whose tech was already so core to conveying that moment-to-moment narrative when you go to the lengths that naughty dog clearly has in its upcoming remake of that original game in upgrading the last of us tech in nearly every conceivable way you can't help but fundamentally alter how a player experiences that narrative and look before we go further i understand the very prevalent backlash that this game's eye-watering price tag of 65 quid has garnered i really do further encouraged by developer remarks stating that the gameplay of the original would remain untouched leading many to question in what world that price point is justified i get it and we'll come on to whether you should take the plunge or not later on there are valid reasons as i see it for you to hold off but i have to be honest it speaks to just how substantial these visual and technical upgrades are something i'm usually not super bothered about that as a longtime fan i feel like despite the developer's comments to the contrary the texture the feel of a game i revisited just two years ago has shifted dramatically i've been noticing things in its story i never picked up on before i've been playing in a noticeably different way whatever your thoughts on the commerce side of things it becomes clear almost immediately that the last of us part 1 is a changed entity one that to me as someone who'd bear in mind didn't have to pay for a copy playstation sent me review code i still found to be a remarkably exciting prospect and of course the most obvious place to start with this is in those drastically upgraded facial animations because let me tell you if this stuff was industry leading before here it's reached another level entirely like look i have no problem with video game violence but if you thought that sarah scene was hard to watch before now it's downright distressing the generally pained expression found in the original morphing into all these panicked little looks and twitches as this little girl fails to make sense of what's just happened to her rendering that blank look teardrop and all that suddenly greets her anguished dad all the more stark to paraphrase what my pal jordan medler at vgc said to me it turns out watching the life drain from a wee lassie's eyes in this level of excruciating detail is about as far from pleasant as one can get or tell anyone's eyes for that matter stealth killing an enemy infected or otherwise now treats you to a smorgasbord of truly harrowing facial expressions as they desperately painfully grasp for air until that moment where they just give up those eyes that were previously darting around as sarah's were convincingly searching for any way out just roll to the back of their head as they slump to the ground all rendered infidelity that goes beyond that of even the last of us part two in silky smooth 60 frames per second in all seriousness the last of us was already a game that forced you into uncomfortable proximity with the brutality of this world and the intimacy of violence that resulted from that but whereas before i merely found it interesting how the game incentivized engaging in this violence in killing rather than just sneaking past encounters you know an empty battlefield lets you freely scout for much needed supplies here i found myself actively fighting that mechanical incentive i was all too glad to stealth my way past as many enemies as i could even if it meant less crafting materials down the line because i just didn't want to have to witness this stuff any more than i absolutely had to this is what i mean when i say the visual upgrades change the very nature of the game you know you're pressing the same buttons in order to make your character do ostensibly the same things they did nine years ago but i'd be willing to bet the effect it has on most players will have changed considerably i mean even pressing those same buttons is itself functionally distinct from the original thanks to the controller you're pressing them on indeed aside from its world-class animations the implementation of the dual sense here is immediately top tier with maybe the best use of the adaptive triggers seen in any game amongst sony's lineup fully utilizing the device's potential as a tonal and storytelling aid in the way i've previously discussed my desire for at length see besides the impressive directionality of haptic feedback felt across the controller where say sarah nuzzling into a couch sends a subtle enough buzz through the handles that you get an actual physical feeling of pre-outbreak comfort that's about to be aggressively replaced with the violent judders of explosions and gunshots the triggers during said gunfights require legitimate effort to pool as if you were holding firing a real heavy piece of machinery it's to the point that in amongst the still utterly thrilling chaos of enemies rapidly flanking my position as my best laid stealth plans inevitably went to pot i found early on that i couldn't make certain shots because i wasn't pulling down hard enough in amongst the frenzy and panic the dual sense doesn't let you dilly-dally through encounters humming and hawing about whether you should take your chance on that shot as if you could just effortlessly pile more into that enemy's general direction should you miss instead the game now demands a kind of ruthless confidence in your decision making it demands you inhabit the role of this man hardened by 20 years of cold brutal struggle speaking of which though it really bears repeating just how crucial the rejigged visuals are in conveying that struggle for example in the original i always found it a bit odd how little joel seemed to have aged in the two decades separating the sarah incident from where we later find him perhaps a comment on how stressful life always was as a single dad trying to make ends meet i guess here though you see every one of those 20 years in his updated appearance his hair is comparatively ashen the fatigue from decades of implied horror etched into his facial features like the rings of a particularly gnarled oak tree joel has seen some [ __ ] in his time and the game's heightened attention to detail ensures that even just by glancing at him this is now an inescapable fact but that's the thing too while i typically associate the detail of the original game's animation focused storytelling with this kind of deep longing or abject sadness you know the aforementioned watch scenes getting you to ask all these questions about how that trauma still hangs like a tarpaulin and the mental gymnastics joel is having to do in the moment to hide the effect it's clearly still having on him the increase in fidelity means that this sadness is far from the only emotion being captured through a character's face now it's in that specific eye twitch of exasperation when someone tells you a piece of information you just can't be arsed hearing the slight stretching of the skin as joel exhaustedly rubs his face it's a character becoming convincingly distracted by something in the distance while they're talking to you it's those tantalisingly brief moments where ellie will crack a shite joke and in amongst the barbarity of day-to-day survival joel will allow the slightest of smirks to crack through his lips before quickly returning to cold dad stoicism it's become cliche over the years to say that a game looks borderline photorealistic but i can't remember a title that comes close to matching these immeasurably subtle movements and the sheer volume of information they convey it legitimately looks like real people acting now with all the humanity and disquieting horror that that brings i mean thanks to the new visuals begging me to just stop and stare at everything around me it brought a whole new significance to the game's environment like it only really struck me this playthrough that you enter into the gallery part of the history museum through some kind of art restoration facility underneath all these works left in various stages of completion and preservation left to rot once more as the length of time they've been lying there the elements they've been exposed to as nature starts to claw itself back against civilization all becomes extremely apparent previously i treated areas like this as mere corridors now they're lived in places each room its own character with its own past life all of this stuff is absolutely incredible and yet if what i've talked about so far regarding technology and storytelling doesn't fill you with the same kind of excitement it does me i think you are completely justified in your desire to skip this we live in a world ravaged by ongoing worsening economic instability the price of a ps5 is increasing in many regions and here we have a remake of an already remastered game that isn't even a decade old yet asking for 65 quid that is a lot of money for a game that a hasn't been out long enough to get most people wistfully looking back and clamoring for a more modern way to play and b is still readily accessible still incredible on multiple different systems like i say i get it but as a long time fan i see this remake as more of a boutique product intended for the most hardcore of hardcore fans or people just getting into the game for the first time who want the absolute best experience out of it the last of us remains an utterly vital masterpiece as far as i'm concerned and if you can justify the steep cost of entry part one is now the definitive way to play it thank you so much for watching be sure to like and subscribe check out my podcast and follow me on twitch if you're so inclined these videos are made possible by the support of my wonderful patrons whose names you're seeing on screen now they've also been getting access to more 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and charlie yang and with that this has been another episode of writing on games thank you all so much for watching stay safe and i will see you all next time
