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.