hello everyone in this video we are going to go through some common relationships of events so first of all we will use a diagram a pictorial representation of events and manipulations with events
is obtained by using a Venn diagram okay here in the diagram the sample space S is represented by the interior of a rectangle and any event say event A is
represented by the interior of a circle since any event must be a subset of a sample space so the circle represented of an event
should be entire
ly within the rectangle let's consider an experiment of rolling
a die once so let A be the event that the outcome is an even number and let B be the event that the outcome is an odd number so the sample space consists of the number one two three four five and six and then event A consists of the sample points with numbers 2 4 and 6 and event B consists of the numbers 1 3 and 5. let's construct a Venn diagram showing event A and event B so we'll use a rectangle to denote the sample space so all
the numbers from 1 to 6 should be included in this rectangle and then for each event we will use a
circle for its representation so here we have two events so we use two circles
one for event A and one for event B and then we will start writing the
numbers okay into the respective circles or the rectangle okay so the number one is in event B so the number one is within Circle B and then the number two appears in event A
so the number two is written within Circle A and then number three appea
rs in
event B so number three is in Circle B similarly number four is in circle A number
five in Circle B and number six in circle A so this is the complete Venn diagram for this example let's take a look at another example okay now let event A to be the event
that the outcome is at most five and B be the event that the outcome is an odd number okay so write out all the possible outcome in the sample space as well as the corresponding outcome in event A and event B so A is the event that the
outcome is at most five so the event A consists of the number one two three four and five and event B is the outcome corresponds to an odd number so event B consists of the number one three and five construct the Venn diagram okay the number one is both in event A and event B so this number one should appear in Circle A and in Circle B and then the number two only appears in event A so the number two is in circle A only and then the number three appears
in both Circle A and Circle B the number
four in circle A and the number five in Circle A and Circle B as well not to forget the remaining number the number six
number six is not in event A nor it is in event B okay so this number six should be outside Circle A and outside Circle B so we have the number six here and then from this Venn diagram you can see some duplication the number one number three and number five are both in Circle A and Circle B so in fact we can combine them to reduce such duplication we can construct the Venn dia
gram like this so here the number one three and five are
still in Circle B and when we consider event A okay the circle A here consists of the number two and four but not to forget that the number one three and five is also in circle A so here Circle A consists of the number one two three four and five and Circle B consists of the
number one three and five and the number six here is outside Circle A and Circle B and this is the complete representation for this example let's move on to the next
one okay so here we let event A to be the
event that the outcome is at most four and B be the event that the outcome is an odd number okay write out the sample space the corresponding sample points for event A and event B so event A consists of the numbers one two three and four and event B consists of the number one three and five construct the Venn diagram the number one appears in both event A and event B
so the number one is in Circle A and Circle B the number two in circle A the number t
hree appears in both Circle A and Circle B the number four in circle A number five in Circle B and then number six is outside Circle A and outside Circle B again we have duplication combine
them to reduce such duplications okay so only the number one and number
three appears in both Circle A and Circle B so the Circle B is not entirely within Circle
A the two circles overlap with each other okay so this is the complete Venn diagram for this example so far we have gone through some
common rel
ationships of events as shown on this slide so in the first Venn
diagram where the two circle has no overlapping region so here we say that event A and event B cannot occur simultaneously in the second diagram Circle B is entirely in circle A so here
event A and event B can occur simultaneously and all sample points in event B are
also the sample points in event A so we can see that when event B occur event A must occur but not the other way around and then in the last diagram the two circles
overlap as well so we can say that event A and event B can occur simultaneously and in this last Venn diagram at least one
sample point is in both Circle A and Circle B next we will be going through
some symbols okay corresponding to the relationship of two events first we will talk about the union of two events let's take a look at the definition the union of two events A and B is denoted by this notation A and then Union B indicate the set of all outcomes that are
included in either A or i
n B or in both events so consider the Venn diagram on the right hand side the colored region corresponding
to the union of event A and event B let's take a look at an example okay
roll a die let A be the event that the outcome is almost four and let B be the
event that the outcome is an odd number so we have the following Venn diagram
showing the relationship of event A event B so the set of A union B consists of
the number one two three four and five okay so in circle A we have the number
one two three and four and in Circle B we have the numbers one three and five so these are the numbers to be included in the set of A union B therefore the set of A union B consists
of the number one two three four and five and the number of elements in this set is equal to five next we move on to the intersection of two events so let's take a look at its definition the intersection of two events A and B denoted by this notation okay read as A intersect B is the set of all outcomes that are inc
luded in both events A and B okay so in the diagram on the right hand side the colored region corresponds to the intersection of event A and event B okay recall the intersection of two events is the set of all outcomes that are included in both events A and B so only the overlapping region corresponds to the intersection of two events in this example okay so let's take a look at an experiment of rolling a die once okay so A be the event the outcome is at most four and B be the event that the out
come is an odd number okay so we have the Venn diagram so what is the set of the intersection of
event A and event B only the numbers included in circle A as well as in Circle B will be the
intersection of the two events A and B so here the set of A intersect B consists of number one and three only and the number of elements in this intersection is two only the number one and three are included in both Circle A and Circle B so the number one and three are the only elements in the intersection
of A and B the next definition we are going to
talk about is the complement of an event the component of event A it is denoted by A dash is defined to be the event consisting of all sample points that are not in A okay so
here is the Venn diagram so I use the circle represent event A and any sample points outside event A is the complement of a denoted by A dash the shaded region okay so let's take a
look at an example suppose you flip a coin okay so your flip a coin the complement of
the eve
nt of getting a head okay suppose the event is to get a head so the complement of the event get a head is the event not getting a head okay so this is the same as to get a tail or another experiment you roll a six-sided die the complement of the event obtain the number five is the event not getting the number five okay so these are two simple examples
showing the complement of event so let's take a look at an example okay
roll a die once so let A be the event that the outcome is almost four an
d let B be the event that the outcome is an odd number so what is the complement of
event A and event B respectively the complement of event A so let's
take a look at what is inside Circle A the number one two three and four are in
circle A so the number five and six are outside Circle A so the complement of
A consists of sample points five and six and the number of elements in the complement of event A is two how about the complement of event B in the circle of B we can see the number 1
3
and 5. the numbers 2 4 and 6 are outside Circle B so the complement of B consists
of sample points two four and six and the number of elements regarding
the complement of event B is three next we move on to the definition of mutually exclusive events so what is the definition when event A and event B have no outcomes in common they are said to be mutually exclusive or we use the term disjoint okay no outcome in common so when we draw the Venn diagram the two circles do not overlap so here ther
e's no overlapping region of
Circle A and Circle B so when there's no overlapping we say that the two events are mutually exclusive or they are disjoint events so the two events cannot occur simultaneously so that A intersect B is an empty set no sample points in common so the number of elements in the
intersection of event A and event B is zero okay so far we have talked about the relationship of two events and we can extend it to more than two events say three or more events here okay so I h
ave event A event B event C and event D okay so let's do some practices okay
so I have a highlighted region here so this highlighted region consists of sample points that are in the complement of event A outside Circle A the sample points in event B the sample point not in Circle C so the
sample point in the complement of event C and also this region is in Circle D okay so what is the corresponding notation corresponds to this region so this is the intersection of the complement of event A
ev
ent B the complement of event C and event D if we know the relationship in
the Venn diagram we can simplify the notation here this region can be
written as the intersection of B and D okay so let's take a look at another region okay the region in the middle of the circle A B and C okay so here this region consists of sample points in event A event B and event C but outside event D so the corresponding notation would be the intersection of A B and C and the complement of event D but we can writ
e it as the intersection of A B and C as it is shown on the slide okay another region okay this portion
okay you can verify that the notation could be written as the intersection of the
complement of event A event B and event C and then the highlighted region here the larger portion so the notation corresponds to the
intersection of the complement of event A event B the complement of
event C and the complement of event D and then we can also confirm that event
A and event D are mutually exc
lusive Circle A and Circle D has no overlapping region we can also say the same for event C and event D as well they are mutually exclusive there's no overlapping region between Circle C and Circle D okay next we will move on to the definition
of collectively exhaustive events so let's take a look at the definition a set of events say event A1 event A2 up to event An is said to be collectively exhaustive
if one of the events must occur so that is the sample space S corresponds to the union of
A1 A2 up to An okay so let's take a look at this
Venn diagram with sample space S1 okay so we have three circles in the diagram A1 A2 and A3 respectively okay so A1 A2 and A3 are not collectively exhaustive because we can have a sample point outside all these three circles if we define one more event say A4 then we can say that A1 A2 A3 A4 are collectively exhaustive at least one of this event must occur from an experiment okay similarly for the sample space S2 A1 A2 A3 are not collectively exh
austive only when we difine A4 where A4 corresponds to the region outside A1 A2 and A3 then we can say that A1 A2 A3 and A4 are collectively exhaustive the differences between S1 and S2 sample space S1 and sample space S2 is that in sample space S1 A1 A2 and A3 are not mutually exclusive and in sample space S2 A1 A2 and A3 mutually exclusive let's take a look at sample space S3 instead of using circles I use some weird shape okay for an event A2 and then the triangle for event A1 and A3
okay so
here A1 A2 and A3 cover all the region of the sample space S3 here so we can say that A1 A2 and A3 are collectively exhaustive events and similarly for sample space S4 A1
A2 A3 are collectively exhaustive event and in sample space S3 A1 A2
A3 are not mutually exclusive and in S4 A1 A2 and A3 are mutually exclusive okay so let's take a look at an example
okay suppose we flip a coin two times so I'll define four events so let event F
to be the event of getting at most one tail event G to be t
he event of getting two faces that are the same event J corresponds to the event of getting a head on the first flip
followed by a head or a tail on the second flip and then the last event event K corresponds to the event of getting all tails okay so we write out the sample points in all the four events so the sample space here consists of sample points HH HT TH and TT where H denotes head and T denotes tails so F event F consists of the event HH HT and TH okay getting at most one tail so eithe
r you can HH no tail or HT or TH and then event G is the event of getting two faces that are the same so the corresponding sample points would be HH and TT and then event J corresponds to the sample points HH and HT so here the first flip is a head and then the second flip is either a head or a tail and then event K is the event of getting all
tails so the sample point in event K is TT Okay so let's take a look at the event and
then compare the event whether the events are mutually exclusive o
r collectively exhaustive okay so let's take a look at events F and G so are event F and event G mutually exclusive are event F and G collectively exhaustive okay so let's take a look at the event F and event G event F consists of sample points HH HT and TH
and event G consists of sample points HH and TT to determine two events whether
they are mutually exclusive then we check whether there is or there are any sample points in common so here the sample point HH appears in both event F and even
t G so these two events are not mutually exclusive are the two events collectively exhaustive okay so we check whether event F as well as event G consists of all the sample points in the sample space okay so HH HT TH and then event G we have the sample point TT so combining event F and event G all the sample points in the sample space appears in either event F event G or both okay so here event F
and G is said to be collectively exhaustive how about event F and event J comparing event F and eve
nt J there is
one sample point HH as well as HT in common so these two events are not mutually exclusive are they collectively exhaustive okay so the sample space consists of the sample point HH which is included in both event F and event J the sample point HT in event F and event J the sample point TH is in event F and then the last sample point in the sample space TT is not in event F and this sample point is not in event J neither okay so
F and J are not collectively exhaustive okay how abo
ut event F and K event F and K are mutually exclusive because there is no sample point in common and then these two events are also collectively exhaustive combining event F and event K all the sample pints in the sample space is
either in event F event K or in both events and then lastly compare the event J and event K event J and event K are mutually exclusive no sample point in common and event
J and event K are not collectively exhaustive because the sample point TH is not in eventually J
it is not in event K okay so in this video we have go through lots of definitions regarding the relationship of events thank you for watching and in the next video we will be talking about some Concepts in probability
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