Conditional probability with sample a sample space
okay Ethan this video is for you and this is also for whoever loves probability including myself and
here's the question we have two tires and we are going to grow both of them and we see that
one of them to four and watch one to find out the probability than the other one it's a six before
I show you guys the solution let me just go for a common mistake and I will also make a remark
probability is pretty hard and it's pretty tricky buy so much fun anyway here we go okay let me just
write down P for probability and open parenthesis and I will describe how much into two inside
of the parentheses right here I said one it's a probability that the add the one it's a six maybe
while just trying to find out the probability that the first one it's a six right so I'll get
a six I won't or maybe the second one could be a six as well okay so it seems like it and I am not
really sure what does this have to do but that seems just how D is right the first one to six
or the sake points are six yeah okay now let's see what's the probability that when you wrote
die and you get six a die has six sides and six is just one of the sides right so it's one side out
of six types so of course it's just 1 over 6 and then we can do the same thing for the second one
and we add them up one over six plus one of the six we get one-third okay so let me tell you guys
that this right here it's not correct let me just go over why first of all you should be suspecting
that this is not correct because we didn't use this information at all and in fact this right
here does affect the probability right and the second thing is that one-third it's not even
the correct answer to this question okay suppose we did have this information at all suppose we
are just trying to find out the probability that one of the ties it's a six namely the first one
to six with a second one to six this right here it's an eiephant occur answer to this and before
I do this let me work this stuff for you guys on the side and in fact we need that right so
let me put that down right here so let's find out the probability that one of the ties it's
a six name D maybe the first one it's a six or the second one it's six and this right here it
was again we don't know this right we didn't have this we didn't know about the four okay now this
right here it's a worse situation and both events in fact can happen at the same time isn't it we
can have the first one is a six and the SEC one also being a six in this case be sure we do some
subtraction be sure we avoid the double counting situation so to work this out what we do is we
have to first figure out the probability that the first one it's a six and then we add it with the
second one to be or six right the probability for that and we have to subtract the probability that
the first one to six and the second one it's also a six and I will just write down both equal to six
so in fact this ratio was wrong because we didn't do this part but once again this and that they are
totally two different question but now you'll see why we need this data by anyway now let's see the
probability that the first one to six is one over six and then this one is one over 6 and then the
probability that both dies come out to PS 6 is the first one to 6 which is with the probability
1 over 6 and you multiply by the probability the second one it's also 6 which is 1 over 6 as well
and now you can just work this out you end up it's 11 over 36 and this right here will be the answer
for this when we don't know about the four at all right okay so that's joke courier this question up
about the one sir hmm so now what exactly does this have to do with the whole question one of
that being for perhaps I wasn't sure yet another common mistake right so let me just write this
down okay now if you really want to use the four somewhere and also the six maybe we're trying to
find out one of them in turf war and the second one it's a six so let me put this down here so
I do have the four in the six huh okay now let's see how we can figure this out well when we roll
two dice the total possible outcome is 6 times 6 namely 36 okay when we draw two dice the total
possible outcome is 36 and I want one of them to be a four and the other one to be a six how many
possible ways can happen well just imagine that we have two dice this one this one maybe I can give
four and six like this or I can get six more like that so we have two possible ways so the answer
to this part is 2 over 36 okay and in my opinion from work doing probabilities it's actually better
sometimes to not reduce the fraction so that's how to use for now all right but this right here once
again it's not the true answer to the original question that's why I happy doing the math in
blue now here is the deal this right here it's called the conditional probability when we see
this happens in fact it narrows things down more so you'll see the actual answer that we have
it's much bigger it's not much bigger the actual answer that we have right here for this question
it's actually bigger than this answer all right so how do we do this let me write this down for you
guys I will first approach it with the formula way and a second D I will show you guys which
the sample space way so this is called conditional probability here's the deal first it's a notation
we'd write it down as probability of a given B this vertical line just means given so sometimes
once you know certain things already in fact you get to narrow things down right and given B here
is the thing right here we see that one of them it's a four so that's the given information right
and we're trying to find out the probability that the other one it's a six that's the a okay okay
so this right here is just a notation and now here is the formula taking just kind of think about it
when we're trying to find out the probability that a given B what we do is we have to first
find out the probability that a and B both happen we are trying to find out the probability that a
and B both happen okay and then we divided by the probability of just B and because of this right
here because of this right here you know you may be able to increase the probability I'll show you
guys why so now do you just do the connection that why we did this right here okay now let
me just write this down we are trying to find out the probability that one of them it's a six right
up back okay so we have the given information and this is the B and on the top is 2a so what
we're trying to do is on the top we have to first find out the probability that one of the time it's
a 6 and the other one it's a 4 and that's exactly this right here so I'll put this down 6 of course
you can switch this to doesn't really matter right okay to the 6 here to the four it doesn't matter
and for the denominator we're trying to get the probability that one of them it's a 4 so on the
bottom years probability okay now in fact we have all the information already what's the top the
probability that one of the ends of what and the other one says it this right here it's exactly
da which is 2 over 36 so they me just write this down and then now let's look at the denominator
what's the probability that one of the dies it's a 4 well let's look back to here earlier this right
here was trying to calculate the probability that one of them it's a 6 and now we're trying to do
it with 4 and do you see the 6 and the 4 matter no I can choose Phi I can choose to I can choose
3 it doesn't matter the probability that one of them is a 4 is the same as a the first one it's a
for war ii winds of war and that's exactly what we talked about earlier we had to first figure
out the probability that the first one is 4 and then we added with the probability that second
one it's a 4 and then we have to subtract both of them be equal to 4 for this we get 1 over 6
for this we also get one of us aches both of them being of work means we have to do 1 over 6 plus
1 over 6 times 1 over 6 which is still this and we still get that and we still put down
in heaven over 36 who in the end - what we want to do to simplify this little fraction it's a
complex fraction you get 2 over 11 and this right here is the answer to the original question right
okay it's a bonus this is the time I will show you guessed with the sample space you guys are
still watching I appreciate it so much now here is the deal and especially when we first learn
about probabilities it might be easier to just look at the sample space to work with so right
here's the idea when we wrote to dice we have a total of 6 times 6 namely 36 outcomes right
so what we're doing is for example this box here means the first one the one in blue is a 1 and
then the wine R is also 1 so it's 1 1 and you can just pretty much continue this part right
here it means 1 2 and then of course 1 3 and so on alright and now here's the idea right here
when the question says we see one of them it's a 4 maybe we're just talking about this column because
this right here we know one of them has to be a 4 right so you have this six possible boxes to
look at but don't forget we might also have this right here because we are also get a 4 so in fact
based on this information we narrowed down to we narrow our simple space down to just these boxes
that I color in black and we have a total of 1 2 3 4 5 6 7 8 9 10 11 do not double count this
box twice this is why we have to subtract the probability that both of them beam before right
and that that will come this twice we have total of 11 of these boxes now - oh that's our guess
this right here the probability that one obtains a 4 it's 11 out of 36 all right and let's see we
are trying to find out the probability that one of them it's six well which part is one of them
it's a six this one four and six right here right so the blue one is four and the red one six and
maybe this one the blue one two six and the gray one it's a four you see we have one two two boxes
out of eleven just like this so by using this simple space like this once you know one of the
bits of war you narrow down your sample space to this 11 boxes only and then you look for the
other one being a six which you have these two possible boxes to look at two over eleven and
that's the answer right here well if you didn't have this information if you're just trying to
find out the probability that one of them is of war and the other ones are six you have to use
the whole sample space because you didn't have that so it's 2 out of 36 which is the one no it
did but anyway hopefully you guys like this video I certainly have a lot of fun discuss probability
just like this and if you guys have more questions which probability was inside that let me know I
will try to answer it but yeah hopefully gets like this video and if you guys are new to my
channel please subscribe I to do like fun math videos for you guys and as always that's it