"Mutually Exclusive" and "Independent" Events (...are VERY different things!)
Summary
TLDRIn 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
- ๐ The video discusses the difference between mutually exclusive and independent events, using examples to clarify the concepts.
- ๐ฏ 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.
- ๐ The probability of mutually exclusive events occurring together is zero, which is exemplified by the students' sports participation survey.
- ๐ 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.
- ๐งฎ The intersection of independent events must be non-zero, as seen with the overlap of students playing basketball and studying modern history.
- ๐๏ธโโ๏ธ An example of dependent events is given where the probability of playing basketball changes when considering students who are taller than 175 centimeters.
- ๐ The video emphasizes that mutually exclusive events are, by definition, dependent, as they cannot occur together, contrasting with independent events.
- ๐ The speaker, Justin, uses numerical examples from a survey of 100 high school students to illustrate the concepts of mutual exclusivity and independence.
- ๐ก The importance of looking at actual data to determine if events are independent is highlighted, as assumptions alone are not sufficient.
- ๐ 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
๐ 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.
๐ 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.
Mindmap
Keywords
๐กMutually Exclusive Events
๐กIndependent Events
๐กProbability
๐กIntersection
๐กConditional Probability
๐กDependent Events
๐กNon-Conditional Probability
๐กHigh School Students
๐กSummer Sports
๐กModern History
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]
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