Probability3

Summary

In this video by Erin Heerey, she explores the concept of combining events as sets in probability. Using examples like rolling dice and determining the sex of kittens, she explains the calculation of probabilities for various events, emphasizing key ideas such as independent events, logical operators like AND, OR, and NOT, and the significance of probability distributions. Additionally, she discusses complementary events and their probabilities, as well as introduces the concept of contingency tables for further exploration in probability.

Highlights

• Understanding combined events as sets helps comprehend probability in games like Settlers of Catan. 🎲
• When rolling two dice, each die rolls independently with six possible outcomes each, forming 36 total combinations. 🎲
• Key logical operators such as AND, OR, and NOT aid in crafting compound probabilities and understanding outcomes. ✅
• Addition rules and complementary events are fundamental skills for mastering probability calculations. ➕
• Using Poppy the cat's kittens as examples simplifies the concept of probability distributions in real-life contexts. 🐱

Key Takeaways

• Combining events as sets enhances understanding of probability, crucial for games like Settlers of Catan. 🎲
• While rolling two six-sided dice, each die and its outcomes remain independent. 🎯
• Logical operators (AND, OR, NOT) help combine events and predict outcomes. ✅
• Mastering probability includes knowing addition rules and complementary events. ➕
• Probability distributions, like predicting kitten genders, simplify complex scenarios. 🐱

Overview

In this intriguing segment by Erin Heerey, the complexity of probability unfolds through relatable examples, such as rolling dice for Settlers of Catan or predicting the gender of kittens. By visualizing events as sets or combinations, Heerey equips her audience with the foundational understanding necessary to navigate more advanced statistical concepts.

The lesson takes an engaging twist with an exploration of logical operators, highlighting their roles in determining the chances of linked events. From there, Heerey introduces crucial rules, like the addition rule in probabilities, providing insightful ways to handle combined event calculations. The video also delves into complementary probabilities, fostering a well-rounded grasp of the subject.

Heerey concludes with a practical illustration of probability distributions, applying the theory to the real-life example of her cat, Poppy's offspring. This brings an approachable and entertaining dimension to probability calculations, bridging theoretical ideas with tangible, everyday situations. An introduction to contingency tables teases further exploration in future segments.

Chapters

• 00:00 - 01:00: Introduction to Combining Events The chapter introduces the concept of combining events by describing events as sets of items grouped together. It starts with a simple example involving rolling a pair of fair six-sided dice, referencing the board game Settlers of Catan where such dice rolling is a common occurrence. The focus is on understanding how to think about and represent events in a combinatory manner.
• 01:00 - 02:00: Rolling Two Six-Sided Dice The chapter titled 'Rolling Two Six-Sided Dice' discusses the possible outcomes when rolling two six-sided dice. It explains that there are 36 possible outcomes, each with an equal probability. The chapter details the combinations of results from each die roll, highlighting that both dice can roll numbers from one to six. It also illustrates specific examples, such as both dice landing on a one or different combinations like one and two, or one and three. The events occurring on die one are distinguished by being listed in green, while those on die two are listed in blue.
• 02:00 - 03:30: Dice Sums and Sample Space The chapter 'Dice Sums and Sample Space' explains the concept of sample space in the context of rolling two six-sided dice. The discussion involves understanding the possible outcomes that can arise from such a roll. The speaker explains how each die has six faces, which means there are six potential outcomes for each die when rolled. Thus, the possible outcomes for a pair of dice rolls are explored. Specific combinations, such as getting a one on the first die and a particular number on the second die, are detailed to illustrate how sample space is constructed. The chapter emphasizes that combinations like getting a one-one or six-six occur in only one way each, highlighting unique outcomes within the sample space.
• 03:30 - 05:00: Logical Operators: AND, OR, NOT The chapter introduces the concept of logical operators using the example of events as sets. It specifically describes an event 'A' involving the sum of two dice equating to three. The possible outcomes to achieve this sum are enumerated as rolling a one and a two, or a two and a one, highlighting the limited possibilities when considering the pair of dice rolls.
• 05:00 - 07:00: Probability Rules and Complementary Events The chapter delves into the concept of probability rules, particularly focusing on complementary events. It illustrates examples using the roll of dice. For instance, to achieve a sum of six on a pair of dice, there are five potential combinations: (5,1), (4,2), (3,3), (2,4), and (1,5). These combinations collectively offer a probability of five out of 36. Additionally, rolling a six on the first die occurs 1/6 of the time, based on the mechanics of dice probability.
• 07:00 - 10:30: Probability Distributions and Example The chapter discusses probability distributions, using the example of rolling two dice. It emphasizes the independence of events, explaining that the outcome of die one does not affect the outcome of die two. Both dice have a 1/6 probability of rolling a six, illustrating the concept of independent probability in discrete events.
• 10:30 - 12:00: Example of Probability in Real Life: Poppy's Kittens The chapter titled "Example of Probability in Real Life: Poppy's Kittens" discusses the concept of events as sets and how they can be combined to understand real-world scenarios. It introduces the use of logical operators such as AND, OR, and NOT to combine these events effectively. The explanation builds on previous lab discussions centered around these operators and their application in probability.

Probability3 Transcription

• 00:00 - 00:30 So now we're going to move into the idea of  combining events. We're going to be talking   about events as sets of items together. So let's  start with a really simple example let's start   with the idea of rolling a pair of fair six-sided  dice. For example, we've talked about that maybe   you play the game Settlers of Catan. In that game  you roll a pair of, hopefully fair six-sided dice.
• 00:30 - 01:00 There were 36 possible outcomes and I've  listed them. They all have an equal probability   so the roll events that occur on die 1 are  listed in green. Events that occur on die   2 are listed in blue. So the possibilities for  die one, you can roll a one through a six. The   same thing is true with die two (a one through  a six). It's possible for the the first die to   land on a one and the second die to land on a one.  You can get a one and a two, or a one and a three,
• 01:00 - 01:30 or a one and a four, or one and a five and  so forth. You could also get a two and a one,   or a three and a one, or a four and a  one, or a five and a one, and so forth.   There is only one possible combination that gives  you one and one and only one possible combination   that gives you six and six, and all of the rest of  these. It's one possible combination because die   one has six choices and die 2 also has six choices  - there are six possible outcomes for each of them   so that's our sample space when we're thinking  about a roll of two, or a pair, of six-sided dice.
• 01:30 - 02:00 Let's think about now events as sets. So one  event, 'A', is that the dice sum to three.   So there were two Elementary outcomes that we  could get that sum to three, a one and a two   or a two and a one. And those are the only two  possibilities where the role of that pair of dice
• 02:00 - 02:30 adds up to three. So that's 2 and 36 or 1 and 18,  if we carry the math out. What's the possibility   that the dice sum to six? You can get a five and  a one, a four and a two, a three and a three,   a two and a four, and a one and a five. So there  are five possible elementary outcomes here that   would meet the criteria of the dice summing to  six, or the numbers on the faces adding up to six,   so five out of 36 chances. If die 1 rolls a  six, that happens six times out of 36, or 1/6
• 02:30 - 03:00 of the time. Die two rolls a six - now remember  the roll of die one and the roll of die two,   even though they're cast at the same time, these  probabilities are independent of one another.   Whether if die one rolls a six, it is totally  independent of whether die 2 also rolls a six.   So Die 2 also has a 6 in 36 or  1/6 chance of rolling a six.
• 03:00 - 03:30 Now events are useful when we think about  them as sets because we can combine them to   understand the world. We talked about some  operators in the lab in a previous week.   so we can think about using what we call 'logical  operators' to combine events as sets. There   are three logical operators that I'd like to  introduce you to today they are AND, OR and NOT
• 03:30 - 04:00 so given the set of event X and Event Y we can  use these logical operators to make new events.   So one event is the is Event X AND Event Y so  both events might occur. We can also say Event   X OR Event Y. That means event X could occur OR  Event Y could occur OR both events could occur.   We can also say NOT event X - event X does  not occur. What's the probability of picking
• 04:00 - 04:30 a not-green marble, which I know it was phrased a  bit oddly when you first were introduced to that,   that's why we did that for conjunctions with  OR, AND and NOT. Let's think about those for   a minute so let's think about event C OR  event D from the previous two slides ago.   Probability of event C, which is die 1  rolls a six is highlighted here in light
• 04:30 - 05:00 grey. it happens six times out of 36. The  probability of event D, die two rolls a six,   also happens six times out of 36. They both have a  one in six chance individually. The probability of   event C is one and six; the  probability of event D is one and six. You can see how these work together. Now  if we wanted to use the AND conjunction
• 05:00 - 05:30 event C AND event D, it's the probability  of event C plus the probability of event D,   minus the probability of their  joint occurrence which is 1 in 36. And the reason [for the subtraction] is we  can't double count that space. So both event   C and event D, both of these have an occurrence  that happens, that's part of that same sample
• 05:30 - 06:00 space. And we can't double count it because that  inflates the probability more than it should be.   So that brings us to what we call the addition  rule. So the probability of event one OR event two   equals the probability of event 1  plus the probability of event 2,   minus the probability of event one AND event two. so we set the probability of event  C was 6 out of 36 D was 6 out of 36.
• 06:00 - 06:30 the probability of event C AND event D was 1  in 36. So the probability of C OR D is six   plus six minus one or 11 out of 36. Now,  if events C and D were disjoint, they had no   overlap, the probability of them occurring at  the same time [were 0], meaning this AND conjunction   cannot occur. If that is equal to zero, then we  can simplify this formula and it becomes just
• 06:30 - 07:00 the probability of event 1 plus the probability  of event 2. So that's how we add probabilities. We can also talk about events that  are complementary to one another.   A complementary event is the event  that covers or completes the rest of   the sample space from what was predicted  or what occurred. This is a concept that's going   to come up later in the course when we  start talking about hypothesis testing.
• 07:00 - 07:30 If we think about a coin flip of heads, the  complementary event to our coin flip of heads is   a coin flip of not heads. The complementary event  to a die roll of three, is a die roll of not-three.   So in this case, the complementary event  to a coin flip of heads is tails and the   complementary event to a die roll of three is  a die roll of one or two or four or five or six.   This s a little  illustration of complementary events.
• 07:30 - 08:00 And that brings us to the subtraction  rule. The probability of not   seeing event one, is one (or one hundred percent)  minus the probability of event one. So if the   probability of rolling a three is one out of six,  the probability of not rolling a three is one of six, minus one or five in six.
• 08:00 - 08:30 Now probability distributions are  lists of all possible events [in the sample space], and   the probabilities with which they occur. Probability distributions can be  really interesting and we use them a   lot in statistics. We'll talk a little bit  about more about that at the end of this   lecture. but before we do that I'm going  to introduce you to someone. This is Poppy.   we adopted Poppy a couple of years ago. And when we  adopted Poppy we thought that we were adopting a
• 08:30 - 09:00 kitty who was spayed (Poppy's a girl). Not only  was she not spayed Poppy was having kittens!!   I'm sure that they knew about that when we  adopted her. Poppy's a lovely cat, and she's no   longer having any kittens, but we can think about  Poppy's kittens as a probability distribution.   The sex of a single kitten has a 50% likelihood  to be male and a 50% likelihood to be female. So if   we think about the probability distribution of the  sexes of Poppy's kittens, we can conceptualize this
• 09:00 - 09:30 based on what the probability is that  that each kitten will be male or female. We   can outline our sample space and get our  probability distribution. So these events, at least   in this case, are "disjoint". The sex of a single  kitten is not both male AND female so these   are disjoint. the probability of a single kitten's sex,  cannot be both. It can be one or the other and each
• 09:30 - 10:00 of the events in a probability distribution  must have a probability between zero and one.   In this case it's 50/50 odds so the probabilities  of the values in the distribution must sum to one.   So if we think about the sex of the first  two kittens [as a probability distribution], well we could get the first   kitten being male and the second kitten  also being male. We could get a male and   a female. We could get a female and a male, or  we could get a female/female combination. And,
• 10:00 - 10:30 each one of those, because the sexes happen with 50  probability, they're the odds, and there are four   events here, their odds actually turn out to be  even, meaning they're equally likely events to occur.   Once the first kitten is born though it  changes the probabilities of getting any   of the other kitten combos right? So if the first kitten...  So before we have any kittens that are born we
• 10:30 - 11:00 can have a 25 chance of getting each of these  possible outcomes for the first two kittens. So we can have a male and a male that's 25 percent  male and a female 25 and so forth. Each one of   these things is equally likely. But, the minute the  first kitten is born and we figure out what it's   got under its skirt, then that changes  the probability of what could happen. So let's
• 11:00 - 11:30 say that the first, actually, the first kitten was indeed  male. So let's not just say, it's actually true.   So Poppy's first kitten was male  and so that means that the probability   of this event and this event [the two female first events] dropped to zero. Because the first kitten   is male now, the second kitten it's the  probability that it is male or female is   totally independent of the probability  of the first kitten, and so now we have   the probability of this kitten or this kitten  being born since the first one was definitely
• 11:30 - 12:00 male. The second one, it turned out, was female this [Male / Female] is the event that happened. And this is what happens   when we're looking at probability distributions.  So we look at how often each set of events   might occur, and what the probability is, and  how that's distributed across a sample space. The next item we're going to talk  about is a thing called contingency   tables, and we're going to talk about  that in the next part of the lecture.