Probability2

Estimated read time: 1:20

Summary

In this discussion, Erin Heerey delves into expanding sample space using examples of coin flips and dice rolls. They explore the probability concepts by calculating probabilities of specific outcomes—such as getting certain combinations of heads in coin flips or rolling a specific number with a die—and discuss mutually exclusive and non-disjoint events. The video also touches on weighted dice, probability distribution in a bag of marbles, and understanding certainty and impossibility in probability. Additionally, graphical representations of probabilities like pie charts and stacked bar plots are explained to visualize data distribution effectively.

Highlights

Flipping three fair coins results in eight possible outcomes. 🎲
Probability of exact outcomes calculated by ratio of matching events to total events. 🔢
Mutual exclusivity in events: can't flip heads and tails at the same time. 🚫
Graphing probabilities visually explained with pie charts and bar charts. 📊
Example of a weighted die illustrating non-equal probabilities. 🎲

Key Takeaways

Understanding sample space: Flipping three coins increases possibilities to eight outcomes! 🎲
Probability calculation described: Exact match events vs. total possible events! 🔢
Mutually exclusive events are ones where neither outcome can happen simultaneously! 🚫
Possibility of belief checked: Even with disbelief in aliens, the probability of their existence isn't negative. 👽
Graphical probability representations: Pie charts and stacked bars make visualization clear! 📊

Overview

Erin Heerey explains the concept of expanding a sample space using the example of flipping three fair coins simultaneously. This multiplies the eventual outcomes to eight distinct possibilities compared to just two outcomes when flipping a single coin. Heerey describes probability through clear examples, walking through the step-by-step calculation of getting exact combinations of heads.

The discussion also covers rolling a fair six-sided die, exploring the probabilities of rolling specific numbers, like six or five, and weighing them against combinations of numbers such as six or five altogether. Heerey emphasizes understanding mutually exclusive events, using dice rolls and coin flips to highlight these occurrences, and elaborates on possible non-disjoint events that can happen simultaneously or singly but not both.

Graphical representations of probability are discussed as well, with pie and stacked bar charts serving as useful tools for visualizing data distribution. Heerey concludes by comparing politicians' truthfulness, allowing for a real-world application of data representation, before wrapping up with a teaser for the next video section.

Chapters

00:00 - 01:30: Expanding the Sample Space with Coins The chapter discusses the concept of expanding the sample space in probability, using the example of flipping multiple coins. It begins with the familiar scenario of flipping a single coin and proceeds to explore what happens when three coins are flipped simultaneously. Each coin is assumed to be fair, meaning each has an equal probability of landing on heads or tails. The full sample space, representing all possible outcomes of the coin flips, is illustrated with a sample space diagram enclosed in curly braces. The chapter emphasizes understanding the complete set of possible outcomes when dealing with multiple probabilistic events.
01:30 - 04:00: Probability of Outcomes with Dice The chapter discusses the concept of sample space in probability with a focus on using coins as an example. It explains how each coin flip represents binary outcomes, either heads or tails, leading to an expansion of the sample space as more coins are introduced. For instance, flipping three coins results in eight possible outcomes, thus illustrating how the complexity of the sample space increases with the number of coins. This sets a foundation for understanding probability distribution with multiple objects like dice.
04:00 - 07:00: Probability in a Bag of Marbles Experiment This chapter explores the concept of probability through the example of a Bag of Marbles experiment. It begins by discussing possible outcomes in a random experiment, highlighting how the sample space grows with each repetition. The focus is on calculating the probability of a specific event, like getting exactly two heads in a coin toss, by considering the number of successful outcomes over the total number of outcomes. The discussion uses simple experiments like coin tosses to explain the broader principles of probability.
07:00 - 10:00: Introduction to Probability Concepts In the chapter titled 'Introduction to Probability Concepts', the concept of probability is explained by examining events within a sample space. The transcript illustrates probability calculation by considering the occurrence of a specific event—in this case, obtaining double-heads in eight possible outcomes. It highlights that the probability of getting exactly two heads is determined by dividing the number of times double-heads appear (three times) by the total number of possible outcomes (eight). Thus, this example establishes a fundamental understanding of how probability is quantified by using a simple event occurrence scenario.
10:00 - 12:00: Graphical Representation of Probabilities The chapter titled 'Graphical Representation of Probabilities' discusses the concept of probability through the example of coin tosses and dice rolls. It explains that the probability of getting exactly two heads in multiple coin tosses is calculated based on favorable outcomes over possible outcomes. Similarly, the probability of rolling a specific number on a fair six-sided die, such as a six, is straightforward as one in six. Moreover, the chapter highlights scenarios involving 'or' probability, such as the likelihood of rolling a six or a five, which involves adding the probabilities of each individual event.

Probability2 Transcription

00:00 - 00:30 Let's talk about how we expand the sample space. Now we've been talking about flipping one coin. Let's think about what happens when when we flip three coins, all of which are fair, at the same time. So what you can see is the sample space diagram here. It's enclosed in the curly braces that I showed you before. So what can happen? What's the full sample space? Coin one can either land heads or tails. Coin
00:30 - 01:00 two can either land heads or tails. And coin three can either land heads or tails. So the sample space expands. All three coins could land heads. We could get heads, heads, tails; we could get heads, tails, heads; we could get heads, tails, tails; tails, heads, heads; tails, heads, tails; tails tails, heads; and all three tails. So now what you can see, is with three coins, the sample space has increased
01:00 - 01:30 from two possible outcomes, to a total of one, two, three, four, five, six, seven, eight possible outcomes. So when we were thinking about the sample space, we're thinking about the number of times a particular random experiment is repeated and the possible outcomes. What's the probability of getting exactly two heads? Now remember we said that the probability of of a particular event occurring, in this case exactly two heads, was equal to the number of
01:30 - 02:00 times that that event occurs in the sample space, divided by the total number of items in the sample space. So we just said there were eight items in the sample space. Double-heads occurs once, twice, three times. So the probability of getting exactly two heads is out of eight possible outcomes, we have double heads occurring three times so there are three outcomes
02:00 - 02:30 that satisfy our condition. So the probability of getting exactly two heads is three of eight. Let's think about another random experiment, the role of a fair six-sided die. What's the probability of rolling a six? well we know what our sample space, is it's this set of numbers. The probability of rolling a six, is one in six. What about the probability of rolling a six OR a five? Well there are two possible outcomes that meet our criteria, six or five,
02:30 - 03:00 right here out of a total of six elements in the sample space so that's two out of six or one-third. How about the probability of getting an even number? Well even numbers occur one, two, three times out of six, so the probability of rolling an even number is 0.5. What about the probability of rolling a five and a six? Well, five and six are mutually exclusive. they can't both occur at the same time. The die does not come up on a side where both of those values are present.
03:00 - 03:30 So with a single roll of a fair six-sided die, we cannot get a six and a five at the same time. they are 'mutually exclusive' events so they can't occur at the same time. This probability is zero out of six, or zero. How about the probability of getting a seven? Also zero, as you might imagine. seven is not on this die. So events can be mutually exclusive, and when they are
03:30 - 04:00 we call them 'disjoint' these are events that cannot both occur together. we cannot flip both heads and tails if we make a single coin flip - in the same flip we can't get both values. You cannot pass and fail the same class in the same term. 'Non-disjoint' events are events that can occur together so you can get an A in, say, 2811 and an A in another class in the same term.
04:00 - 04:30 You could also flip two coins at the same time. That could generate one heads and one tails. So sometimes events can be non-disjoint, meaning they can occur together. Now, do the probabilities have to be equal? Well it turns out that sometimes we can flip a loaded die. The sum of the probabilities of the elementary events can never be greater than one or 100 percent, but the probabilities don't have to be equal. So if we have a weighted die where
04:30 - 05:00 probability of rolling a one is 25 percent and all the rest are the same what we then do is we take 1 minus 25 that gives us a total of 75 left and we divide it by the total number of other outcomes that gives us 75 or 0.75 divided by 5 or 0.15. If the probability of rolling a one is 25, because this is a weighted die or an unfair die,
05:00 - 05:30 the probability of rolling any of these other numbers is lower. It's now not even odds anymore it's less. The odds of these other rolls are less. Here's another random experiment. Imagine you have a bag of marbles. This isn't a very nice looking bag but there you go. The probability of drawing a purple marble from a bag with three purple marbles, four red marbles, three blue, and two green. What's the probability of picking a purple marble? So our
05:30 - 06:00 sample space includes all of the possible choices. Here there are three purples, four reds, three blues and two greens. These are sort of color-coded here. There are a total of 12 marbles in the bag. There were three purple marbles so the probability of picking a purple marble is three in twelve or one quarter (25 percent). How about picking a marble that is not green? Well, there are two green marbles so there are ten not-green marbles. So any one
06:00 - 06:30 of these is a marble that is not green. So 10 out of 12 times or five out of six (83 percent) of the time you will pick a marble that is not green in this particular bag of marbles. And now you can start to see how we compare these probabilities to one another. When we're comparing probabilities we can think about the probability of some event being
06:30 - 07:00 zero or zero percent. That event is impossible (probabilities cannot be negative). So it's probably impossible for me to be abducted by aliens on my way home. I don't believe in aliens, now they might exist, but I don't believe in them. I believe that the probability of me being abducted by aliens on my way home from work today is zero percent. This is an impossible probability. Other events
07:00 - 07:30 are totally certain. We all know the old adage the only things in life that are certain are death and taxes. It is a hundred percent certain that at some point in your life you will pay taxes. and It is also 100 certain that at some point in your life, you will meet your end; that event is certain. An event that has a probability of 0.5 is an event that is as likely to happen as not to happen. So if there's a 50 chance of rain today, that means it is as likely to rain as it is to not rain.
07:30 - 08:00 An event that has a probability of 25% or 0.25 is an event that is less likely to not happen than to happen. so if you think about a 25% chance of rain, odds are it's not going to rain; A 75% chance event is more likely to happen than it is to not happen. So when we're thinking about comparing these probabilities, we can now take that probability weighting that we've calculated and guess how likely a particular event is to happen.
08:00 - 08:30 So, on the previous slide we talked about that bag of marbles and we talked about picking a not-green marble. The probability of picking a not-green marble was 0.83; 83 percent. so it was more likely that we would pick a not- green marble than that we would pick a green one. If we're representing these things graphically, one of the common representations for probabilities is
08:30 - 09:00 a pie chart. And this is a nice pie chart here. This is, it's an old one it shows desktop browser market share as of 2016. And there are a couple of things you can take away here. The first one is that these probabilities that are listed in the in the cells here, or in the slices of pie, sum to 100 percent. And the sizes of these pieces of pie are proportional to the proportion of the pie that
09:00 - 09:30 they're worth. So Chrome in 2016 had a market share of 61.2; so 61.2 percent of people were using Chrome as their web browser. 12.1 percent were using Internet Explorer; 15.5 using Firefox; and 11.3 percent were using something else like Safari, or Opera, or Edge, or something else entirely. The sum of the proportions is a hundred percent. We can also think about a stacked
09:30 - 10:00 bar chart. This is another one of these interesting displays. This is a great one that we saw a few years back. It was it's a comparison of politicians telling, well making statements that were then evaluated by an organization called PolitiFact, which is basically an independent fact checker. So it graded a whole bunch of political statements as true, mostly true, half true, mostly false, false, or liar liar pants on fire,
10:00 - 10:30 and then it assigned these values a proportion of statements attributed to each politician that fell into each one of these categories. What you can see is that politicians like Donald Trump spent a lot of time saying things that are rather on the not so good side of true and they tell very little truth. Politicians like the former president Obama
10:30 - 11:00 of the United States, spent a lot more time telling things that were true and at least half true than he did telling people falsehoods, we can also change what these stacked bar plots look like a little bit. This is a diverging stacked bar plot and it's another good representation of a dichotomous variable. This is a variable that has a positive side and a negative side so unlike the
11:00 - 11:30 the pie chart that we saw on the previous slide when we were talking about web browsers, that one is not representing a divergent kind of variable where there aren't multiple categories per individual, you either are mostly using one web browser or you're mostly using another but people are they're asked about their primary web browser rather than about what how often they use other web browsers. In fact that was not generated based on people asking
11:30 - 12:00 at all it was, generated based on hits to Google. So this one here though, because each person has statements that can fall into multiple categories, this is a really nice representation. It's aligned along a baseline of zero and it organizes these bars based on the proportion of of the bar that falls into each one of these categories. So this is a stacked bar chart that is aligned against a
12:00 - 12:30 vertical baseline, and you can see it in comparison over here where you can see these bands getting wider, kind of in this in a diagonal pattern. Here, you can see them all nicely aligned and you can see where the truths and half truths and mostly false and our pants on fire statements fall. Very interesting. So I'm going to stop there and we will move into the next section in the next video.