Probability Part 1: Rules and Patterns: Crash Course Statistics #13

Summary

In this Crash Course episode, Adriene Hill explores the concept of probability, distinguishing between empirical and theoretical probabilities. Empirical probability is derived from observed data, such as the ratio of girls in families, while theoretical probability represents an ideal truth. Hill introduces the addition rule for calculating the probability of multiple, non-mutually exclusive events and the multiplication rule for independent events. She also discusses conditional probabilities, using examples like medical screenings and the likelihood of simultaneous events, such as Cole Sprouse's presence and 'Free Ice Cream Night' at IHOP, to illustrate these principles.

Takeaways

• 🧠 The human brain's tendency to see patterns, such as faces in inanimate objects, is known as pareidolia.
• 📊 Probability in statistics is defined in two ways: empirical probability, which is observed in data, and theoretical probability, which is an ideal or universal truth.
• 🔍 Empirical probability is derived from actual data and provides an estimate of the theoretical probability but is subject to uncertainty and randomness.
• 🎰 Theoretical probability is an idealized concept that can be estimated through empirical observations, such as the probability of winning a jackpot on a slot machine.
• 🍭 The addition rule of probability is used to calculate the probability of one or more events occurring, considering whether events are mutually exclusive or not.
• 📝 Probability notation, like P(Red) for the probability of drawing a red Skittle, simplifies the expression of complex probability scenarios.
• 🎲 When events are not mutually exclusive, the full addition rule accounts for the overlap by adding the probabilities of individual events and subtracting the probability of their joint occurrence.
• 🍨 The multiplication rule is used to find the probability of multiple independent events occurring together, such as Cole Sprouse being at IHOP and it being 'Free Ice Cream Night'.
• 🏥 Conditional probabilities, like P(Cancer | Positive Test), are crucial in medical screenings to understand the likelihood of a condition given a test result.
• 🎟️ Probability can inform decision-making in everyday life, such as choosing between movie tickets or applying to multiple colleges, by helping to set realistic expectations.

Q & A

What is pareidolia and how does it relate to the human brain's pattern recognition?

Pareidolia is a psychological phenomenon where the brain perceives a familiar pattern, such as a face, in an unrelated object. It occurs because the human brain is exceptionally adept at recognizing patterns, sometimes to the extent of seeing them even when they aren't present.

What are the two types of probability discussed in the script?

The two types of probability discussed are empirical probability and theoretical probability. Empirical probability is derived from observed data, while theoretical probability represents an ideal or universal truth that we cannot directly observe.

How is empirical probability different from theoretical probability?

Empirical probability is based on observed data and carries some uncertainty due to being a sample of all available data. In contrast, theoretical probability is an idealized, exact value that represents the true likelihood of an event occurring in the universe.

What is the addition rule of probability, and when is it used?

The addition rule of probability is used to calculate the probability of either one event or another occurring. It is used when events are mutually exclusive, meaning they cannot happen at the same time. The rule states that the probability of A or B occurring is P(A) + P(B).

What is meant by mutually exclusive events in the context of probability?

Mutually exclusive events are those that cannot occur simultaneously. In probability, the probability of two mutually exclusive events happening at the same time is zero.

How do you calculate the probability of getting a red or purple Skittle from a bag if each color has a 20% chance?

The probability of drawing a red or purple Skittle is calculated by adding the individual probabilities of each color since they are mutually exclusive. So, P(Red or Purple) = P(Red) + P(Purple) = 0.2 + 0.2 = 0.4 or 40%.

What is the full addition rule for probability, and when is it used?

The full addition rule for probability is used when events are not mutually exclusive. It is calculated as P(A or B) = P(A) + P(B) - P(A and B). This rule accounts for the probability of both events occurring together to avoid double counting.

What is the multiplication rule in probability, and how is it used?

The multiplication rule is used to calculate the probability of two or more independent events occurring together. It is expressed as P(A and B) = P(A) * P(B), provided that the events are independent, meaning the occurrence of one does not affect the probability of the other.

What is the difference between independent and dependent events in probability?

Independent events are those where the occurrence of one does not affect the probability of the other. Dependent events, on the other hand, have probabilities that are influenced by whether the other event has occurred.

How are conditional probabilities used in medical screenings, as mentioned in the script?

Conditional probabilities are used in medical screenings to determine the likelihood of having a condition, given a positive test result. For example, P(Cancer | Positive Test) represents the probability of actually having cancer given that the test result is positive.

Why are probabilities important in everyday decision-making?

Probabilities are important in everyday decision-making because they help set realistic expectations and guide choices by estimating the likelihood of different outcomes. They can inform decisions such as planning events, making backups, and assessing risks.

Outlines

00:00

🧠 Understanding Probability and Pattern Recognition

This paragraph introduces the concept of probability through the lens of pattern recognition, specifically pareidolia, where the brain perceives familiar patterns like faces in random objects. It sets the stage for discussing empirical and theoretical probabilities. Empirical probability is derived from observed data and is subject to uncertainty, whereas theoretical probability represents an ideal or universal truth. The example of a slot machine illustrates how empirical probability can estimate theoretical probability. The paragraph also introduces the addition rule of probability for mutually exclusive events, using the selection of Skittles as an example, and explains the use of probability notation, such as P(Red) for the probability of drawing a red Skittle.

05:04

🎰 Probabilities of Independent and Dependent Events

The second paragraph delves into the addition rule for non-mutually exclusive events, using the example of rolling a die and flipping a coin. It explains the need to adjust the addition rule by including the probability of both events occurring together to avoid double counting. The concept of conditional probability is introduced, showing how it can be used to calculate the likelihood of one event given the occurrence of another. The paragraph uses a hypothetical scenario involving actor Cole Sprouse and a 'Free Ice Cream Night' at IHOP to illustrate the multiplication rule and the calculation of probabilities for independent events. It also touches on the importance of conditional probabilities in medical screening, using cervical cancer screening as an example to demonstrate how false positives and negatives affect the interpretation of test results.

10:05

🎬 Applying Probability to Everyday Life

The final paragraph emphasizes the practical application of probability in everyday decision-making. It discusses how understanding probabilities can help in setting realistic expectations and making informed choices, such as choosing movie tickets or applying to colleges. The paragraph also highlights the unpredictability of certain events, like catching red lights, and suggests that while probabilities can't always provide exact outcomes, they are a valuable tool for navigating the uncertainties of life. The host encourages viewers to consider probability as a part of their decision-making process, even when faced with the complexities of real-world scenarios.

Keywords

💡Pareidolia

Pareidolia is a psychological phenomenon where the mind perceives familiar patterns or images, such as faces, where they do not exist. In the context of the video, pareidolia is used to introduce the concept of pattern recognition, which is a fundamental aspect of how our brains process information and is directly related to the study of statistics and probability. The video uses the example of seeing faces in inanimate objects like bell peppers to illustrate how our brains are wired to find patterns, even when they are not present.

💡Probability

Probability is a measure of the likelihood that a particular event will occur. It is a central concept in statistics and is used to make inferences and predictions based on data. The video defines probability and distinguishes between empirical and theoretical probability. Empirical probability is based on observed data, while theoretical probability represents an ideal or true value that may not be directly observable. The video uses the example of playing a slot machine to explain how empirical probability can be estimated from the frequency of an event occurring.

💡Empirical Probability

Empirical probability is the probability of an event based on observed data. It is an estimate derived from a sample of the population and is subject to uncertainty due to the randomness and limited size of the sample. In the video, the ratio of girls in individual families is given as an example of empirical probability, which provides an insight into the true theoretical probability but may not be exactly equal to it.

💡Theoretical Probability

Theoretical probability refers to the true probability of an event occurring, which is considered an ideal or absolute value that exists independently of any particular sample or observation. The video explains that theoretical probability is not directly observable but can be estimated through empirical observations, such as the long-term frequency of winning a jackpot in a slot machine game.

💡Addition Rule of Probability

The addition rule of probability is a principle used to calculate the probability of either one event or another (or both) occurring. It is particularly useful when dealing with non-mutually exclusive events. The video explains the rule using the example of selecting a Skittle, where the probability of picking a red or purple Skittle is the sum of their individual probabilities. The rule is adjusted for non-mutually exclusive events by adding the probability of both events occurring together.

💡Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. In probability, if two events are mutually exclusive, the probability of both occurring together is zero. The video uses the colors of Skittles as an example, where a Skittle cannot be red and purple at the same time, making these color choices mutually exclusive.

💡Multiplication Rule

The multiplication rule is used to calculate the probability of two or more independent events occurring together. It is based on the idea that the probability of multiple events occurring is the product of their individual probabilities. The video illustrates this with a scenario involving the chance of actor Cole Sprouse being at a local IHOP and it being 'Free Ice Cream Night' on the same day.

💡Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred. It is denoted as P(Event 1 | Event 2) and is calculated by dividing the probability of both events occurring by the probability of the second event. The video discusses the importance of conditional probabilities in medical screenings, such as determining the likelihood of having cervical cancer given a positive test result.

💡Independence

In probability, two events are considered independent if the occurrence of one event does not affect the probability of the other event occurring. The video explains this concept using the example of Cole Sprouse's presence at IHOP and 'Free Ice Cream Night' being independent events, meaning that the probability of Cole being there is not influenced by whether it is 'Free Ice Cream Night' or not.

💡False Positives and False Negatives

False positives and false negatives are terms used in the context of diagnostic testing. A false positive occurs when a test indicates that a condition is present when it is not, while a false negative indicates that a condition is absent when it is actually present. The video discusses these concepts in the context of cervical cancer screenings, explaining how understanding these probabilities can influence medical recommendations and patient expectations.

Highlights

Pareidolia is the phenomenon where our brains perceive patterns, like faces, in non-face objects.

Probability is introduced as a measure of the likelihood of events occurring.

There are two types of probability: empirical, based on observed data, and theoretical, an idealized concept.

Empirical probability is derived from the ratio of specific outcomes in observed data, such as the number of girls in individual families.

Theoretical probability represents an ideal or universal truth about the likelihood of an event.

The empirical probability can estimate the theoretical probability but is not always equal due to sample randomness.

An example of estimating probability is given by playing a slot machine and calculating the win rate.

The addition rule of probability is explained for calculating the probability of one event or another occurring.

Mutually exclusive events are defined as events that cannot occur at the same time, such as selecting a red or purple Skittle.

Probability notation, such as P(Red), is introduced to simplify the expression of probability calculations.

The probability of getting a red or purple Skittle is calculated using the addition rule, resulting in a 40% chance.

The full addition rule is presented, which accounts for the probability of two non-mutually exclusive events occurring together.

The multiplication rule for probability is introduced to calculate the likelihood of multiple independent events happening simultaneously.

An example using the multiplication rule involves calculating the probability of Cole Sprouse being at IHOP on 'Free Ice Cream Night'.

The concept of independent events is explained, where the occurrence of one event does not affect the probability of another.

Conditional probabilities are discussed, showing the probability of one event given the occurrence of another.

The practical application of conditional probabilities in medical screenings, such as for cervical cancer, is examined.

The importance of understanding probabilities in everyday life decisions, such as planning a night out with friends, is highlighted.

The video concludes by emphasizing the usefulness of probability in setting realistic expectations and making informed decisions.

Transcripts

Hi, I’m Adriene Hill, and Welcome back to Crash Course, Statistics.

If you’ve ever seen a face in an onion, or a grilled cheese, or any other inanimate

object, you’ve experienced pareidolia, which is a product of our brains that causes us

to see the pattern of a face in non-face objects.

This happens because our brains are so good at seeing patterns, that they sometimes see

them when they’re not really there, like a face in this bell pepper.

And faces aren’t the only patterns we see.

Our brains recognize patterns in everything, especially in sequences of events, like the

kind we’ll see today as we start talk about Probability.

INTRO

Alright, first let’s just establish a more specific definition of what probability is,

because the way we use the word in everyday life can be different from how we use it in Statistics.

Statisticians talk about two types of probability: empirical, and theoretical.

Empirical probability is something we observe in actual data, like the ratio of girls in

each individual family.

It has some uncertainty, because like the samples in experiments, it’s just a small

amount of the data that is available.

Empirical probabilities, like sample statistics, give us a glimpse at the true theoretical

probability, but they won’t always be equal to it because of the uncertainty and randomness

of any sample.

The theoretical probability on the other hand, is more of an ideal or a truth out there in

the universe that we can’t directly see.

Just like we use samples of data to guess what the true mean or standard deviation of

the population is, we can use a sample of data to guess what the true probability of

an event is.

Say you play a slot machine over and over, you’ll be able to guess the probability

of winning the jackpot by counting the number of times you win, and dividing it by the number

of times you played.

If you play 100 times and win 6 times, you can be pretty sure that the probability of getting

a jackpot is around 6/100 or 6%.

Now this isn’t to say that you can rule out that the true probability is 5% or even

10%, but you’re relatively sure that it’s close to 6%, and not, say 99%.

So, the empirical probability can be a good estimation of the theoretical one, even if

it’s not exact.

So far we’ve been talking about the probability of just one event, but often there may be

two or more events that we want to consider, like what if you want to know the probability

of picking a purple OR a red skittle from a bag.

The proportion of each color in a bag of Skittles is roughly equal, 20% for each of the 5 colors.

So let’s say you randomly select a Skittle without looking.

For this, we need the addition rule of probability.

Since a Skittle can’t be two different colors at once, the color possibilities are Mutually Exclusive.

That means the probability of a Skittle being red AND purple at the same time is 0.

So, we can use the simplified addition rule which says that the probability of getting

a Red or Purple Skittle is the sum of the probability of getting a Red, and the probability

of getting a purple.

Since we’re going to be talking a lot about probability in the next few episodes I’m

going to introduce a little notation.

Instead of writing out “the probability of Red” we can use the notation P(Red).

The probability of getting a red OR purple would then be written P(Red or Purple).

So far we know what the probability of Red or Purple is, P(Red) + P(Purple), or 0.2+0.2

That equals 0.40 or 40%.

I like all skittles so the probability that I will get a skittle I like is 0.2 + 0.2 +0.2

+0.2 +0.2.

That’s 100%.

Good odds!

Red and Purple Skittles are mutually exclusive, but not all the events we’re interested in are.

For example, if you roll a die and flip a coin, the probability of getting a tails is

not mutually exclusive of rolling a 6, since you can both roll a 6 and flip tails in the

same turn.

Since P(tails or 6) ≠ 0, these two events are not mutually exclusive, and we’ll need

to adjust our addition rule accordingly.

The full version of the addition rule states that P(tails or 6) = P(tails) + P(6) - P(tails and 6).

When two things are mutually exclusive, the probability that they happen together is 0,

so we ignored it, but now the probability of both these things happening is not zero,

so we’ll need to calculate it.

You can see here that there are 12 possible outcomes when flipping a coin and rolling a die.

There are 6 outcomes with a tails, and 2 outcomes with a 6.

If we add all of those together we get 8, but by looking through the chart, we can tell

that there are only 7 possible outcomes that have either a tails or a 6.

When we count T’s and 6’s independently, we double count the outcomes that have both

If we didn’t subtract off the probability of (tails and 6), we would double count it.

Let’s put these probabilities into a Venn Diagram we can see even more clearly why we

need to subtract P(Tails and 6).

If this Circle is all the Times we flip tails, and this circle is all the times we roll a

6, this overlapping area is counted twice if we simply added the two circles together.

In this simple case, we could easily see what the probability of tails and 6 is, but sometimes

it’s not so easy to figure out.

That’s why we have the Multiplication Rule, which helps us figure out the probability

of two or more things happening at the same time.

Let’s say you just found out that actor Cole Sprouse goes to your local IHOP pretty

often, and there’s a 20% chance that he’ll be there for dinner any given night.

And yeah, I know that’s not how people work but we’re going to say that’s how Cole

Sprouse works.

Anyway to top that off, your local IHOP has a promotion where they randomly select certain

nights to be “Free Ice Cream Night” in the hopes that customers will keep coming

back in case that night is the night.

Each night there’s a 10% chance that it will be “Free Ice Cream Night”.

Now, you love ice cream and you like like Cole Sprouse--as do we all--and your perfect

night would include them both.

So you try to calculate the probability that will happen on your visit tonight.

Using the multiplication rule, multiply the probability that Cole Sprouse will be at IHOP,

0.2, with the probability that it will be Free Ice Cream Night, 0.1.

And you come to the sad realization that there’s only a 2% chance that you’ll get to see

Cole and get free dessert tonight.

When we want to know the probability of two things happening at the same time, we first

need to look at only the times when one thing--Cole is at IHOP--is true, which is 20% of the time.

Now that we reduced our options to just Cole nights, out of all these Cole times, how often

is it free ice cream time?

Only 10% of Cole nights.

10% of the original 20% leaves only a 2% chance that both will happen at the same time.

But you could always change your expectations and calculate the probability of getting either

by using the addition rule.

Cole or free Ice cream which, is calculated by adding the probability of Cole, to the

probability of Free Ice Cream, minus the probability of both--so we don’t double count anything.

You realize that there’s a 28% chance that something good will happen tonight, so you

decide to still go, no matter what you’re going to get French Toast.

Cole Sprouse and Free Ice Cream Night are independent.

Cole doesn’t have any secret knowledge about when it’s Free Ice Cream Night, so it has

never affected his decision to come.

Two events are considered independent if the probability of one event occurring is not

changed by whether or not the second event occurred.

In more concrete terms, if Cole’s decision to go to IHOP is independent of IHOP’s decision

to give out free ice cream, than the probability of Cole showing up should be the same on both

ice cream and non ice-cream nights, since he’s just choosing randomly.

We write conditional probabilities as P(Event 1 | Event 2).

Conditional probabilities tell us the probability of Event 1, given that Event 2 has already happened.

If two events are independent--like Cole and ice cream night--then we expect P(Cole | Ice Cream Night)

to be the same as just plain ole P(Cole), since the two things are unrelated.

If P(Cole | Ice Cream) wasn’t the same as plain ole P(Cole), then that means that Ice

Cream night might somehow affect Cole’s decision to show up at IHOP.

We calculate conditional probability P( Event 2 | Event 1) by dividing the probability of

Event 1 and Event 2 by the Probability of Event 1.

The role of conditional probabilities are particularly important when we consider medical screenings.

For example, when screening for cervical cancer it used to be recommended that all adult women

get screened once a year.

But sometimes the results of the screenings are wrong.

Either they can say there’s something abnormal when there isn’t (called a false positive)

or that everything is all clear when it’s really not (called a false negative).

This is exactly the kind of scenario where knowing the likelihood that something is actually

abnormal in this case cervical cancer given that you’ve gotten positive tests results

would be useful.

That is P(Cancer | Positive Test).

When looking at the data of people who DON’T have cancer, 3% will get a false positive.

And people who DO have cancer will get false negatives 46% of the time.

This means we miss a lot.

And maybe freak some people who don’t need to be freaked out.

The logic of conditional probabilities can help us make sense of why doctors have recently

recommended that these tests be done less frequently in some cases.

In the United States, the rate of cervical cancer is about 0.0081%, so only about 8 in

100,000 women get cervical cancer.

Using our rates of false negatives and positives, we can see that for every 100,000 women in

the US, only about 4...and we’re rounding here of the about 3,004 people with positive

tests actually had abnormal growths.

That means the conditional probability of having cancer, given that you got a positive

test is only 0.1%. Give or take. We're rounding.

And these positive tests require expensive and invasive follow up tests.

And I just want to point out that conditional probabilities aren’t reciprocal.

That is to say P(Cancer|Pos Test) isn’t the same as P(Pos Test| Cancer) which would be about 50%.

In real life you’re not always going to know the probability of Cole Sprouse showing

up at the iHop--he’s unpredictable that way.

Unpredictable like ...pretty much the rest of life.

It can be very difficult to put a specific probability on a lot of everyday situations.

Like how likely it is that your teacher will call-in sick today.

Like whether or not you’re going to catch all the red lights on your way to school.

Probabilities can--as we’ve seen--require a lot of calculations--and there’ not always

time for that.

But that doesn’t mean they belong only on the school-only side of your brain.

Say you want to go out on a Friday night with friends.

More than anything you don’t want it to suck.

Last week you wound up on the couch watching Sandy Wexler, again.

You know it’ll be hard to get tickets to see Black Panther so you make a backup plan

just in case.

You can always stream Get Out without stealing it of course.

But if you’re determined to see Black Panther in the theater, Probability will help set

your expectations.

If you’ll only settle for center row tickets you’re more likely to be disappointed.

Your chance of seeing Black Panther is going to be greater if you’re willing to settle

for whatever tickets you can get.

Probabilities help us understand why it makes sense to apply to more than one college.

Why we should shouldn’t expect that the first short story you write will be get an

A and be published in the New Yorker.

And how likely it is that you’ll get mono.

Given your significant other has mono.

Probability can help you figure that out too.

Thanks for watching. We'll see you next time.