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Multiplication & Addition Rule - Probability - Mutually Exclusive & Independent Events
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now we're going to talk about two basic
rules of probability the first one is
the probability of forget an event a or
event B can be written as the
probability of a and union with B this
is equal to the probability of event a
no current plus the probability of event
B occurring
minus the probability of event a and
event B occurring together so make sure
you know this now sometimes you may be
dealing with two events that are
mutually exclusive for mutually
exclusive events the probability of get
an event a and B together is zero so
therefore the probability of getting a
or B becomes P of a plus P of B if a and
B are mutually exclusive events but you
can always use this formula which will
always work regardless of what type of
events you're dealing with the next type
so let's say if well first let's talk
about conditional probability the
probability of a given B is the
probability of getting a and B divided
by the probability of getting B so what
this means is that if we multiply both
sides by P of B we can get this equation
the probability of getting a and B is
the product of the probability of
getting a given B times the probability
of event B occurring now what about this
scenario what is the probability of B
given a this is still the probability of
a and B occurring but divided by the
probability of a so if we rearrange this
equation by multiplying both sides by P
of a and we see that the probability of
a and B can also be written as the
probability of B given a time's the
probability of a so notice the
sometimes the two events that you're
dealing with might be independent events
an independent event is an event that
does not depend on another event for
independent events the probability of a
given B is equal to the probability of
event a occurring because a does not
depend on B likewise the probability of
B given a is equal to the probability of
event B occurring because B does not
depend on a so when dealing with
independent events we get this equation
the probability of a and B occurring
is simply the probability of a times not
plus but times the probability of event
B occurring if the events are
independent if not then you can get one
of these equations depending on what you
were given one thing I do want to
mention is that when you see a and B
occurring this means that these two
events are occurring at the same time it
doesn't mean a and then B which could be
different from B and then a so because
the events occur at the same time the
order is not relevant otherwise the
formulas may be affected so just keep
that in mind therefore the probability
of a and B occurring is the same as the
probability of B and a occurring because
these two events occur at the same time
so I just want to add that clarification
for those of you who may have questions
on it now let's go ahead and put this
information to good use Sarah is
deciding which courses she wants to take
in her next college semester the
probability that she enrolls in an
algebra course is 0.3 and the
probability that she enrolls in a
biology course is 0.7 t the probability
that she will enroll in an algebra
course given that she enrolls in a bio
course is 0.4 t so Part A what is the
probability that she will enroll in both
an algebra course and a biology course
so take a minute pause the video and
using the formulas that we discuss go
ahead and get the answers to these questions
questions
so first let's begin by writing down
what we know the probability that she
takes an algebra course that is P of a
is 0.3 zero and the probability that she
takes a bio course that's 0.7 zero the
bility of a given B that is that she
takes an algebra course given that she
takes a bio course that's point four
zero so with this information we can now
focus on Part A so we're looking for the
probability that she will take both
algebra and bio so what's the formula
for this this is equal to the
probability of a given B times the
probability of either B occur so we can
see that P of a given B is 0.4 0 and the
probability that she will take a bio
course is 0.7 0 times 7 is 28 so 0.4
times point 7 is point 28 so there's a
28% chance that she's gonna take both
algebra and biology now let's move on to
Part B what is the probability that she
will enroll in an algebra course or a
biology course so this time we're
looking for P of A or B so what's the
formula for this so this is going to be
based on the addition rule as opposed to
the multiplication rule so this is P of
a plus P of B minus P of a and B now the
probability that event a will occur is
0.3 and a probability that she's gonna
take the biology course is 0.7
the probability that she will take both
algebra and biology we have it here that
is 0.28 so let's go ahead and subtract
these numbers so 0.3 plus 0.7 minus 0.28
this is equal to 0.72 so there's a 72%
chance that she will take algebra or
biology now let's move on to Part C are
the two events independent what would
you say what are the requirements for
two events to be independent in order
for events a and B to be independent the
probability of a given B must be equal
to the probability of a are they equal
the probability of a given B we could
see that it's point four and a
probability of event a occurring is 0.3
so these two are not equal therefore the
two events are not independent of each other
now what about Part D are the two events
in mutually exclusive what would you say
in order for the two events to be
mutually exclusive the probability of
event a occurring and event B occurring
must be equal to zero is this true well
we know that a and B the probability of
these two events occurring is point two
eight because that's what we calculated
in Part A so it does not equal zero
which means that events a and B are not
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