Probability1

Summary

In this engaging lecture about probability, speaker Erin Heerey explores everything from the basic definitions to the fascinating history behind the theory. She delves into how probability plays a crucial role in both psychology and everyday decision-making, covering basic definitions, historical anecdotes such as de Mere's Paradox, and contrasting Frequentist and Bayesian views. Heerey also discusses random processes and experiments, emphasizing the practical application of probability in games of chance, statistics, and real-world scenarios. The lecture reveals how probability helps predict events, the difference between quantitative and subjective certainty, and the challenges of making inferences without experimentation.

Highlights

• Erin Heerey opens the lecture by discussing the role of probability in psychology and how it's applied in everyday situations, like checking the weather. 🌦️
• Yahtzee and dice games serve as classic examples where probability helps determine the likelihood of certain outcomes. 🎲
• An intriguing story about the history of probability involves 'de Mere's Paradox,' a gambling problem discussed by Pascal and Fermat. 🎭
• The concept of random processes is illustrated with examples like coin flips, dice rolls, and everyday activities where outcomes are uncertain. 🤔
• The lecture contrasts two important views on probability: Frequentest, which focuses on long-term frequency, and Bayesian, which incorporates subjective certainty. 🌐
• Practical applications of probability include determining the likelihood of rain, which Heerey illustrates with personal anecdotes about deciding when to carry an umbrella. ☔

Key Takeaways

• Probability is a measure of how likely an event is to happen, and it's essential in various fields including psychology and gaming. 🎲
• The history of probability began with gambling, showcasing its real-world implications right from the start. 🎰
• Blaze Pascal and Pierre Fermat were instrumental in developing the equations that underpin probability theory. 🔢
• Random processes are situations with uncertain outcomes, and they can range from simple coin flips to complex human behaviors. 🔄
• Frequentest and Bayesian are two primary views of probability, offering different perspectives on predicting events. 🧠
• Probability calculations help us make informed decisions, weighing different potential outcomes based on known conditions. ⚖️

Overview

In the world of probability, Erin Heerey takes us on a journey from its ancient roots in gambling to its current applications in psychology and statistics. By defining probability as the likelihood of an event occurring, she sets the stage for exploring its significance in everyday decision-making. Whether predicting weather conditions or analyzing data, probability offers valuable insights into the patterns of random events, often revealing more than meets the eye.

Heerey delves into the history of probability, sharing captivating tales of gambling and paradoxes. We learn about the Chevalier de Mere's favorite bets and how a simple question about dice led to groundbreaking discoveries by the mathematical minds of Blaze Pascal and Pierre Fermat. These stories provide a backdrop for understanding how probability became a vital mathematical discipline, influencing games of chance and scientific inquiry alike.

The lecture also highlights the dual perspectives of probability: Frequentest and Bayesian. Heerey explains how Frequentist views rely on long-term frequencies derived from random experiments, while Bayesian perspectives incorporate personal beliefs and certainties about events. This duality is crucial in real-world applications, where probability helps individuals and scientists alike make sense of uncertainties and plan for future possibilities.

Chapters

• 00:00 - 00:30: Introduction to Probability In the 'Introduction to Probability' chapter, the lecture discusses various elements crucial to understanding probability. It covers the history of probability theory, defines key terms, and explores basic concepts and their applications in psychology, including contingencies.
• 00:30 - 01:00: What is Probability? Probability is a measure of how likely an event is to occur, given the present conditions. For example, determining the likelihood of rain today can be done by checking a weather app for a percentage or observing the sky to make an informal estimate.
• 01:00 - 02:00: Probability in Games and Psychology The chapter delves into the application of probability in everyday decision-making and games, particularly focusing on how people use probability to predict weather outcomes and make decisions like taking an umbrella when it's cloudy. It also explores how probability is crucial in games of chance. Using the example of the dice game Yahtzee, the chapter discusses calculating the probability of rolling specific combinations, like getting five of a kind, highlighting the importance of understanding probability in both daily life and gaming strategies.
• 02:00 - 04:00: Historical Perspective on Probability The chapter begins by highlighting the importance of understanding probability within the realms of statistics and psychology, exemplified by the question of whether observed statistical results are merely coincidental or indicate an underlying phenomenon.
• 04:00 - 06:00: Random Processes and Experiments The chapter explores the concept of random processes and experiments, highlighting probability as a fundamental idea. It questions how frequently events occur by chance, and touches upon the historical context of probability, noting its origins in gambling.
• 06:00 - 09:00: Definitions in Probability The chapter titled "Definitions in Probability" discusses historical treatises on probability, beginning with one called 'On the Art of Dice' by Emperor Claudius of Rome, which is now lost. The chapter then recounts a notable event in 1654, where Chevalier de Mere, a passionate gambler, posed a famous probability problem known as 'The Problem of Dice'.
• 09:00 - 13:00: Frequentist and Bayesian Views This chapter covers the probabilistic concepts from both frequentist and Bayesian perspectives, focusing on a particular gambling scenario. The narrative describes a gambler who regularly bets on rolling at least one six in four rolls of a die, a bet that statistically favors him. The core of the discussion centers around the probability that one six will appear in four rolls, which is calculated to be two-thirds, thus providing favorable odds for the bettor. The chapter showcases the distinction between the frequentist approach, which focuses on the probability based on long-term occurrence without prior beliefs, and the Bayesian approach, which incorporates prior beliefs into probability assessment.
• 13:00 - 15:00: Predicting Random Events with Probability In this chapter, the focus is on an interesting scenario in probability related to predicting the outcome of random events, particularly involving dice rolls. A person tried betting on rolling a double six within 24 rolls of two dice, expecting to win, but surprisingly, he lost more often. This apparent contradiction is explored through what is called 'de Mere's Paradox'. The problem's complexity warranted insight from notable mathematicians of the time, Blaze Pascal and Pierre Fermat, highlighting the intricate nature of probability despite the seemingly straightforward situation.
• 15:00 - 18:00: Calculating Probability The chapter delves into the foundational aspects of probability theory, initiated by the correspondence between notable mathematicians Pascal and Fermat. They established fundamental equations that are central to probability theory today. The discussion begins with definitions, focusing on random processes, which are scenarios where possible outcomes are known based on experiments or observations.
• 18:00 - 20:00: Conclusion and Next Steps In this conclusion chapter, the concept of randomness and its impact on various scenarios is discussed. The chapter emphasizes the unpredictability of outcomes in different situations, such as gambling, stock market fluctuations, and music shuffle playlists. These examples illustrate how random processes affect everyday decisions and scenarios, reinforcing the understanding that while we know the possibilities, the actual outcome remains unknown.

Probability1 Transcription

• 00:00 - 00:30 In this lecture we're going to  be talking about probability. We'll be talking about a couple of important  elements of probability including the history   of probability theory. I'm going to be  walking you through some definitions   and we're also going to be talking about some  basic ideas and ways that we use probability   in the work that we do in Psychology. We're  also going to be talking about contingencies   that's one of those really important ways  in which probability applies to psychology.
• 00:30 - 01:00 So let's start out with 'what is probability'?  Probability refers to how likely it is that a   given event will happen, based on the conditions  that are present. So you might ask for example   what are the chances that it will rain today.  And you can open up your weather app and I'm   sure it will give you a probability of rain.  so that probability might be 30% it might be   50 percent. You might also just look out  your window and look at the sky and think,   "oh there's not a cloud in the sky today  maybe that means the chance of rain is low,"
• 01:00 - 01:30 or you might look out your window and say, "it's  cloudy and grey, I better take my umbrella today." One of the domains in which probability is  extremely useful are in games of chance,   like Yahtzee for example. Yahtzee is a  dice rolling game in which participants or   players try to get five of a kind, along with  a number of other combinations of dice rolls.   How likely is it that you're going to get  five of a kind across the rolls in your turn?
• 01:30 - 02:00 In statistics, and in Psychology, we often  talk about: is this statistical result that   we've observed in our data, could that be a chance  occurrence or is there something going on? I can   also ask the question what are the odds that my  spouse has been abducted by aliens? That sounds   like a silly question, but in fact there's  a proof in what we call Bayesian analysis,   that suggests that the odds that my spouse  has been abducted by aliens go up if he
• 02:00 - 02:30 is both late coming home from work, and  he is not working on a grant right now.   This is the idea of probability: how  often do things happen by chance?   The history of probability is a long one.  In fact the formal study of probability was   initially used for gambling - so this might help  you down at the gambling parlor. The very first
• 02:30 - 03:00 treatise on probability was called 'On the Art of  Dice'. It was written by Emperor Claudius of Rome   and unfortunately, that treatise has been  lost so it's not available in modern times.   What you can read, if you want to, is in 1654 the  Chevalier de Mere, who was an inveterate gambler,   posed 'The Problem of Dice'. Now the history  of his problem of dice was this. He had a
• 03:00 - 03:30 favorite bet, and he used to bet regularly  on this, his favorite bet was getting one 6,   in four rolls of a die. And there he won more  often than he lost. Now a single rule of a single   die has a one in six chance of rolling a six, as  you know, assuming it's a fair die. So in four   rolls, the probability of getting a six is two  in three. And those are pretty favorable odds.
• 03:30 - 04:00 So then he started making a new bet - and  it was rolling one double six in 24 rolls   of two dice. And there he lost more often  than he won. Why could that be? Even though   those bets look like they should be the same,  they're really not. To understand this problem,   de Mere asked Blaze Pascal who was a  friend of his, who in turn shared what   he called 'de Mere's Paradox' with another  famous mathematician named Pierre Fermat.
• 04:00 - 04:30 The theory of probability was the product of the  correspondence between Pascal and Fermat and so   they identified a number of the equations that we  think about when we're thinking about probability.   So let's begin by talking about terms, what  a random process is. So we often talk about   random processes. These are the outcomes of our  experiments. These are the outcomes of all kinds   of things. A random process is a situation  in which we know what outcomes could happen,
• 04:30 - 05:00 but we don't know which specific outcome will  happen. So for example, you might toss a coin,   you might roll a die, you might go to the  casino and make a bet on the roulette wheel.   These are random processes. Whether your stock  portfolio goes up or down tomorrow is a random   process. Here's another good one, that we're  all familiar with, what's the next song in   the iTunes Shuffle? When you put shuffle on, you  know what the playlist is, but you don't know what
• 05:00 - 05:30 the next song is going to be. So these are random  processes. and it turns out that random processes,   the degree to which a process is random, depends  on how you define it. I could ask the question:   how many cups of coffee will you drink tomorrow?  And you might tell me that is entirely your own   free will it's totally up to you and you make  that decision. And sort of you do. But I could   conceptualize that, as the experimenter, I could  define that as a random process. I could say this
• 05:30 - 06:00 is a process that's related to how many hours  of sleep you're going to get tonight, how much   dopamine is floating around in your brain right  now, how many cups of coffee do you usually drink,   do you have an exam or assignment due tomorrow  because if you do, maybe you're going to drink   more coffee. So I can conceptualize that  same question very differently than you.   I can conceptualize it as a random process where  you might not. So as long as it's a situation in
• 06:00 - 06:30 which we know what outcomes could happen but we  don't know which specific outcome will happen,   it can be defined as a random process.  If there's no variance in the outcome   then it's probably not a random process  - then we know what MUST happen. So let's talk about some definitions now.   These are important as we think  about and represent probability.   One of the things that we consider is what we call  a 'random experiment' and that's the process of
• 06:30 - 07:00 observing the outcome of a single chance event  - so noting the outcome of a single coin toss   that would be considered a single chance event. A 'random variable' is a variable that results  from the measurement of your random experiment.   So it can take on a value that describes  the outcome of the random experiment. I   might have a variable, let's call it X,  because we're being very creative today,   X equals heads because heads was what came up on  my coin. I could also have a variable that is the
• 07:00 - 07:30 kind of trial that happens in my experiment.  Now when I do many of the experiments I do,   we do these things online and there are different  trial types and they're randomly mixed up. So I   know what the possible trial types are, but I  don't know what the next trial type will be,   so the trial type might be a  random variable in my experiment. 'Elementary outcomes' are all the possible  results of a random experiment. So when you
• 07:30 - 08:00 are flipping coins, you can either  get heads or you can get tails. There are two possibilities; you can be correct  or incorrect; you can pass or you can fail;   there are all kinds of elementary outcomes out  there that you could consider. The 'sample space'   is the set of all possible elementary outcomes  for your experiment or set of experiments.   So these are often noted as 'sample spaces'  and are often noted with these little curly
• 08:00 - 08:30 braces here, along with a representation of the  possible outcomes inside (e.g., heads or tails). It looks the same in this example  as in the elementary outcomes,   but let me tell you, we'll talk about  this in a little bit, let me tell you   that if we have two coin flips that sample  space is going to become more complicated. And finally, the 'probability' is numerical  weight, that ranges from zero to one,
• 08:30 - 09:00 that describes the likelihood  of an outcome's occurrence   so the probability of heads is 0.5 the  probability of flipping a tails in a   coin flip is also 0.5. If the coin is fair,  both of those outcomes are equally likely to   occur. They don't need to be, but that's  what's that's what typically happens or   that's how we typically conceptualize them.  If the coin is fair, they are equally likely.
• 09:00 - 09:30 So how we understand this probability of an  event is the proportion of times the event   would occur if we conducted an infinite number  of random experiments within the sample space.   So this is the long-term frequency at which  the event occurs over many many many trials.   If we're thinking about flipping a coin, if we're  thinking of dice, if we're thinking about asking   participants to respond to a particular trial  type and then measuring how quickly they respond,
• 09:30 - 10:00 that's the 'Frequentest' view. Now there's  an alternate view of probability called the   'Bayesian' View. In the Bayesian view,  we certainly include this frequent view,   this idea that over time, time will  tell which events occur most frequently. But the Bayesian view also includes the subjective  level of certainty that a person has about an   event. So imagine that we're talking about whether  or not it will rain today, or the likelihood of
• 10:00 - 10:30 rain. Now for the same event you and I could have  very different levels of certainty or different   viewpoints about it. So we could both open our  weather app and it could be the same weather   app. If it's the same weather app, it's going to  tell us the same probability of rain in our area,   assuming we're both in London. So let's say we  open our weather app and it says there's a 35   chance of rain today. Is that enough of a chance  for you to take your umbrella? Hmm. good question.
• 10:30 - 11:00 So I walk to work - I walk every day. I walk about  three kilometers one way, so it's not a long walk   but it's certainly, I don't want to be rained on.  So if I see a 30 or 35 percent chance of rain on   my weather app, I will probably bring my umbrella  with me just in case. You maybe you take the bus   or maybe you drive or maybe you ride your bike or  maybe you just like walking in the rain in which
• 11:00 - 11:30 case a 35 chance of rain is not going to be enough  of a chance to get you to bring your umbrella to   school. Instead you might choose to have a higher  threshold, maybe a 70 chance, before you decide to   bring your umbrella or your rain jacket. So this  is the idea that two people might have really   different levels of certainty or different  viewpoints about the likelihood of an event.   This Bayesian viewpoint is nice especially  when we're talking about things where we
• 11:30 - 12:00 don't have the chance to do experimentation.  What's the probability that global warming is   influenced by human activity? There are a lot of  people out there who believe that that is true,   but there are also a lot of people out there who  believe that it's not. So they do not believe   that global warming is influenced by human  activity and do not feel that their behavior   should change. But we only have one Earth we  can't experiment - we don't have Planet B,
• 12:00 - 12:30 we can't repeat this experiment and so what we  need to do is we need to make our inferences   based on probability, and that becomes difficult  when different people have different ideas or   different viewpoints, different levels  of certainty, about that relationship. So the big question we want to ask is how can we  predict an event that is fundamentally random.   For example, how can we predict a single coin  flip of a fair coin? What's the probability of
• 12:30 - 13:00 getting heads. So the probability of getting heads  in a coin flip is noted as 'P(H)' so it's written,   we see it written as 'P' and then with an  'H' in parentheses here; probability heads.   It's represented by this equation: the  probability of an event is equal to the   number of outcomes that meet a certain  condition, divided by the total number   of equally likely outcomes. so in a coin flip of  a fair coin, heads and tails are equally likely,
• 13:00 - 13:30 so the probability of getting heads there's one  condition one outcome that meets the criterion,   its heads, and there were two possible  outcomes that are equally likely.   so the probability of getting heads is  one divided by 2 or 0.5 so it's one half.   So that's how we represent the  probability of a specific event. So here's another visualization of that. If  we think about a coin flip of a fair coin,
• 13:30 - 14:00 the probability of heads equals, there are two  possible outcomes heads and tails, the sample   space includes one heads one tails, because we're  flipping the coin once, both of these are equally   likely. So now we can calculate the number of  outcomes that meet a condition which is one,   this one right here, heads, divided by the total  number of equally likely outcomes which is two,
• 14:00 - 14:30 and that's fifty percent. So that's  the idea with probability. I'm going   to pause there and we'll pick up in the next  section with how we expand the sample space.