"Mutually Exclusive" and "Independent" Events (...are VERY different things!)

Summary

In this educational video, Justin from zedstatistics.com clarifies the difference between mutually exclusive and independent events, using examples of high school students' sports and study choices. He explains that mutually exclusive events, like playing basketball or cricket, cannot occur simultaneously, ensuring their intersection's probability is zero. In contrast, independent events, such as playing basketball and studying modern history, show no impact of one event on the other, as seen when the probability of playing basketball remains the same regardless of studying history. Justin also points out the common misconception that mutually exclusive events are the same as independent events, emphasizing that they are, in fact, opposites, with the former being a form of complete dependence.

Takeaways

• 🏀 The video discusses the difference between mutually exclusive and independent events, using examples to clarify the concepts.
• 🎯 Mutually exclusive events are defined as events that cannot occur at the same time, such as a student playing basketball or cricket in the summer.
• 📊 The probability of mutually exclusive events occurring together is zero, which is exemplified by the students' sports participation survey.
• 📚 Independence in events is determined by whether the occurrence of one event affects the probability of another, demonstrated by the study of modern history and basketball playing.
• 🧮 The intersection of independent events must be non-zero, as seen with the overlap of students playing basketball and studying modern history.
• 🏋️‍♂️ An example of dependent events is given where the probability of playing basketball changes when considering students who are taller than 175 centimeters.
• 🔄 The video emphasizes that mutually exclusive events are, by definition, dependent, as they cannot occur together, contrasting with independent events.
• 📈 The speaker, Justin, uses numerical examples from a survey of 100 high school students to illustrate the concepts of mutual exclusivity and independence.
• 💡 The importance of looking at actual data to determine if events are independent is highlighted, as assumptions alone are not sufficient.
• 🌐 The video is part of a series on zedstatistics.com, where Justin educates on statistical concepts, and viewers are encouraged to donate to an education charity.

Q & A

What is the main difference between mutually exclusive and independent events?

Mutually exclusive events cannot occur at the same time, meaning the intersection of the two events is zero. Independent events, on the other hand, can occur simultaneously, and the occurrence of one event does not affect the probability of the other.

How many students out of 100 play basketball in the given example?

40 out of 100 students play basketball.

What is the probability of a student playing both basketball and cricket in the example?

The probability is zero, as these are mutually exclusive events.

What is the definition of mutually exclusive events as explained in the video?

Mutually exclusive events are defined as events where the probability of both occurring at the same time is zero.

How many students out of 100 play cricket in the example?

25 out of 100 students play cricket.

What is the probability of a student playing basketball given they study modern history?

The probability remains the same as the probability of playing basketball without considering modern history, which is 0.4 or 40 out of 100.

What does it mean for two events to be independent?

Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.

How many students out of 100 both play basketball and study modern history?

8 out of 100 students both play basketball and study modern history.

What is the probability of a student playing basketball given they are taller than 175 centimeters?

The probability is 0.7, indicating that the events are dependent.

Why are mutually exclusive events considered the most extreme form of dependence?

Mutually exclusive events are considered the most extreme form of dependence because the occurrence of one event completely excludes the possibility of the other event occurring.

What does the speaker do with the funds raised through the super thanks button on YouTube?

The speaker donates all funds raised through the super thanks button to an education charity.

Outlines

00:00

🏀 Understanding Mutually Exclusive Events

This paragraph introduces the concept of mutually exclusive events using the example of high school students choosing to play either basketball or cricket during summer. It explains that these two sports are mutually exclusive because a student cannot play both at the same time. The video presents data showing 40 out of 100 students play basketball, 25 play cricket, and 35 play neither. The intersection of the two events (playing basketball and cricket at the same time) is zero, illustrating the definition of mutually exclusive events. The probability of each event is calculated, and it is emphasized that the impossibility of the intersection (playing both sports) is key to defining mutual exclusivity.

05:00

📚 Exploring Independence in Events

The second paragraph delves into the concept of independent events, contrasting it with mutual exclusivity. It uses the same group of students but changes the second event to studying modern history, which is hypothesized to be unrelated to playing basketball. The paragraph explains that independence is determined by whether the occurrence of one event affects the probability of another. Data is presented showing 8 out of 100 students both play basketball and study modern history, indicating an intersection. The probability of playing basketball is calculated with and without the condition of studying modern history, demonstrating that the latter does not affect the former, thus confirming the independence of the events. The paragraph concludes by contrasting this with a dependent event scenario where the probability of playing basketball changes when considering students taller than 175 centimeters, indicating a dependence.

Keywords

💡Mutually Exclusive Events

Mutually exclusive events are two or more events that cannot occur at the same time. In the context of the video, this concept is illustrated with the example of students choosing to play basketball or cricket. Since a student cannot play both sports during the summer, the probability of playing basketball and cricket simultaneously is zero. This is a key point in understanding that mutually exclusive events have no overlap, and hence, cannot happen together.

💡Independent Events

Independent events are two events where the occurrence of one does not affect the probability of the other. In the video, this is demonstrated by comparing the probability of a student playing basketball with and without the condition of studying modern history. The video explains that the probability remains the same regardless of the condition, indicating that the events are independent.

💡Probability

Probability is a measure of the likelihood that a particular event will occur, often expressed as a number between 0 and 1. In the video, probabilities are calculated for various events, such as students playing basketball or cricket, to illustrate the concepts of mutually exclusive and independent events. For instance, the probability of a student playing basketball is given as 40 out of 100, which is 0.4.

💡Intersection

The intersection of two sets refers to the elements that are common to both sets. In the context of the video, the intersection is used to determine whether events are independent. For mutually exclusive events, the intersection is zero, indicating no common elements. For independent events, there must be an intersection, as seen with students playing basketball and studying modern history.

💡Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has occurred. The video discusses how to calculate the probability of a student playing basketball given that they study modern history. It's used to assess whether the events are independent, as the probability of playing basketball remains the same regardless of whether the student studies modern history.

💡Dependent Events

Dependent events are events where the occurrence of one affects the probability of the other. The video contrasts this with independent events by showing that if a student is taller than 175 centimeters, they are more likely to play basketball. The conditional probability of playing basketball given the student's height is different from the unconditional probability, indicating dependence.

💡Non-Conditional Probability

Non-conditional probability refers to the probability of an event occurring without any conditions being applied. In the video, the non-conditional probability of a student playing basketball is discussed as 0.4, which is then compared to the conditional probability given certain conditions, such as playing cricket, to illustrate the difference between dependent and independent events.

💡High School Students

The video uses a sample of 100 high school students to illustrate the concepts of mutually exclusive and independent events. This sample is used to provide concrete examples and numerical data to support the explanations of statistical concepts, making the abstract ideas more tangible and easier to understand.

💡Summer Sports

Summer sports are used in the video as an example to explain mutually exclusive events. The video mentions basketball and cricket as summer sports that students cannot play simultaneously, which helps to clarify the concept of mutually exclusive events in a relatable context.

💡Modern History

Modern history is used as a subject of study in the video to demonstrate the concept of independent events. The video explains that studying modern history does not affect the probability of a student playing basketball, thus illustrating the independence of these two events.

Highlights

Mutually exclusive and independent events are distinct concepts, often confused as the same.

Mutually exclusive events cannot occur simultaneously, as exemplified by students choosing between basketball and cricket in summer.

The probability of mutually exclusive events occurring together is zero, defining their mutual exclusivity.

In the example, 40 out of 100 students play basketball, and 25 play cricket, with no overlap.

Independence in events is assessed by the lack of influence one event has on the probability of another.

Event A (playing basketball) and Event B (studying modern history) are used to illustrate independence.

Eight students out of 100 both play basketball and study modern history, indicating an intersection.

Independence is confirmed when the probability of playing basketball is not affected by studying modern history.

The definition of independence is presented: the probability of A is equal to the probability of A given B.

An example of dependent events is given with Event B changed to students being taller than 175 centimeters.

The probability of playing basketball given the condition of being tall (Event B) is higher, indicating dependence.

Mutually exclusive events are explained as the most extreme form of dependence, not independence.

The channel's educational charity initiative is mentioned, where donations are directed to support education.

The video concludes with an invitation to visit zedstatistics.com for more videos and to support the channel.

Transcripts

g'day team welcome to this video on

mutually exclusive versus independent

events

my name's Justin and if you want to see

any of my videos you can look up at the

zedstatistics.com

this one's interesting this one comes as

there's been a bit of confusion around

what these two concepts mean and some

people actually think they're the same

thing which they very much are not so

let's explore their differences

diving straight in let's have a look at

mutually exclusive events

now for this we're using a an example of

a hundred high school students that have

been surveyed let's just presume they've

been you know questioned on a variety of

different factors facing their schooling

event a we're defining as a student

playing basketball and event B is a

student electing to play cricket

now because these are both summer Sports

a student can't play both of them at

once so you can see that 40 students out

of 100 play basketball in this example

25 play cricket and 35 play neither of

those two sports summing to a hundred

students

now of course we can find the

probability of each of these two events

pretty simply 40 out of 100 for

basketball 25 out of 100 for Cricket uh

the important thing to note is that the

intersection is actually zero so there

is no possibility of playing both Sports

and this symbol here means the

intersection

so the probability of event a and event

B happening at the same time is zero and

guess what that is the definition of

mutually exclusive

it means that a student electing to play

cricket excludes the possibility of him

playing basketball and vice versa

and vice versa not vice versa

so they're mutually exclusive

let's have a look at Independence now

so we're using the same 100 high school

students here

so event a is still that a student plays

basketball but here we're saying that

event B is a student studying modern

history

now you might think that studying modern

history doesn't really affect the

possibility of someone playing

basketball and you might be right

but we need to have a look at the

numbers and that's the key to

independent events we can't assess

whether these two events are independent

until we actually take note of the

numbers

and you'll note here that there is in

fact a intersection between event A and

B so eight students out of the hundred

both play basketball and study modern

history

and that's important you can't have

independent events without

some intersection

we'll see why that is in a little bit in

just a little bit but for the moment we

have to realize that we're going to do

some calculations here

so the question we're going to ask here

is what is the proportion of students

that are playing basketball

now if we just look at the basketball

set here we know it's still going to be

40 over 100 because we have 32 students

that play basketball without studying

modern history

eight students that play basketball with

studying modern history

so if you just ignore modern history

altogether you still have 40 on 100

playing basketball

now if we apply the condition that a

student studies modern history

we can still try to assess the

probability of playing basketball so

we're just looking now at event B and

within event B it's still going to be

the same ratio of students playing

basketball so 8 out of 20 students play

basketball within that modern history

set right so applying the condition

of studying modern history didn't affect

the probability of playing basketball

so that's what I was saying you can only

assess Independence and indeed these are

independent events you can only assess

that Independence through looking at the

numbers

so if we're looking at the definition of

independence we can say that event A and

B are independent if the probability of

a

is equal to the probability of a given B

so I've put in Brackets here applying

the condition B does not affect the

probability of a or in our case applying

the condition of this student studies

modern history

didn't affect their probability of

playing basketball

all right so what's an example where

there is a dependence or in other words

two events are not independent

well let's take event a again the same

one where a student plays basketball and

let's in let's change event B to where a

student is taller than 175 centimeters

now your gut's probably going to be

telling you that there's going to be

some overlap here and students that are

taller than 175 centimeters are probably

more likely to want to play basketball

for obvious reasons right so

let's have a look and see if these are

in fact independent events

and we can do the same thing we can find

the probability of a which is still 0.4

the probability of a given B here

for that we're just focusing on event B

well there's 35 students that are taller

than 175 centimeters and play basketball

and there's 50 students that are taller

than 175 centimeters all up

so that conditional probability is 0.7

so we can say here that these two events

are in fact not independent

or in other words they are dependent

right

so that's good now here's a really

interesting thing

at the very beginning of this video I

said that people sometimes confuse

mutually exclusive events with

independent events and they conflate

those two concepts they think they're

one and the same

but in reality if something's mutually

exclusive

they have to be very much dependent in

other words not independent right

let's have a look at the example we used

right at the beginning a student playing

basketball versus a student playing

cricket

we know the probability of a student

playing basketball is 40 or 0.4 and if I

asked you

what's the probability of a student

playing basketball

given they play cricket you would tell

me that it's zero right so clearly that

conditional probability is not the same

as the

non-conditional probability of 0.4

so that's why I said right at the

beginning that when we have the

independent events actually need an

intersection to possibly be independent

if there's no intersection you'll have

mutually exclusive events which are the

most extreme form of dependence actually

interesting right

I thought so anyway uh my name's Justin

and all of the videos I put up on

zedstatistics.com if you liked the video

you know what to do but here's an extra

little thing if you want to donate some

money to the channel via the super

thanks button I do I push that money on

to an education charity I don't keep any

of that myself so all funds raised

through the super thanks option that

YouTube have made available will be sent

to my choice in education charity and

you can look in the description to see

what charity I am supporting for this

particular video anyway I'll catch you

next time and see you around

[Music]