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In this problem, we will be
helping Romeo and Juliet meet
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up for a date.
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And in the process, also we'll
review some concepts in basic
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probability theory, including
sample spaces
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and probability laws.
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This problem, the basic setup
is that Romeo and Juliet are
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trying to meet up for a date.
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And let's say they're trying
to meet up for
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lunch tomorrow at noon.
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But they're not necessarily
punctual.
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So they may arrive on time with
a delay of 0, or they may
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actually be up to 1 hour late
and arrive at 1:00 PM.
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So the other thing that we
assume in this problem is that
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all pairs of arrival times-- so
the time that Romeo arrives
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paired with the time they
Juliet arrives--
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all of these pairs are
equally likely.
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And I've put this in quotes,
because we haven't really
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specify exactly what
this means.
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And we'll come back to
that in a little bit.
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The last important thing is that
each person will wait for
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15 minutes for the other
person to arrive.
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If within that 15-minute
window the other person
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doesn't arrive, then they'll
give up and they'll end up not
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meeting up for lunch.
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So to solve this problem, let's
first try to set up a
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sample space and come up with a
probability law to describe
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this scenario.
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And let's actually start
with a simpler
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version of this problem.
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And instead of assuming that
they can arrive at any delay
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between 0 and 1 hour, let's
pretend instead that Romeo and
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Juliet can only arrive in
15-minute increments.
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So Romeo can arrive on time
with a delay 0, or be 15
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minutes late, 30 minutes
late, 45 minutes
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late, or one hour late.
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But none of the other
times are possible.
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And the same thing for Juliet.
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Let's start out with just the
simple case first, because it
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helps us get the intuition for
the problem, and it's an
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easier case to analyze.
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So it's actually easy
to visualize this.
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It's a nice visual tool
to group this sample
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space into a grid.
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So the horizontal axis here
represents the arrival time of
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Romeo, and the vertical
axis represents the
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arrival time of Juliet.
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And so, for example, this point
here would represent
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Romeo arriving 15 minutes
late and Juliet
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arriving 30 minutes late.
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So this is our sample
space now.
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This is our omega.
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And now let's try to assign
a probability law.
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And we'll continue to assume
that all pairs of arrival
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times are equally likely.
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And now we can actually
specifically specify what this
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term means.
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And in particular, we'll be
invoking the discrete uniform
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law, which basically says that
all of these points, which are
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just outcomes in our
probabilistic experiment--
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all of these outcomes
are equally likely.
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And so since there are 25 of
them, each one of these
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outcomes has a probability
of 1 over 25.
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So now we've specified our
sample space and our
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probability law.
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So now let's try to answer
the question, what is the
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probability that Romeo
and Juliet will meet
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up for their date?
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So all that amounts to now is
just identifying which of
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these 25 outcomes results in
Romeo and Juliet arriving
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within 15 minutes
of each other.
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So let's start with this one
that I've picked out.
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If Romeo arrives 15 minutes
late and Juliet arrives 30
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minutes late, then they will
arrive within 15 minutes of
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each other.
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So this outcome does result in
the two of them meeting.
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And so we can actually highlight
all of these.
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And it turns out that these
outcomes that I'm highlighting
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result in the two them arriving
within 15 minutes of
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each other.
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So because each one has a
probability of 1 over 25, all
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we really need to do now
is just count how many
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outcomes there are.
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So there's 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13.
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So the probability in the end
is for the discrete case.
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The discrete case--
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I'm referring to the case where
we simplified it and
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considered only arrival times
with increments of 15 minutes.
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In this case, the probability
is 13 over 25.
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So now we have an idea of how
to solve this problem.
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It amounts to basically coming
up with a sample space, a
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probability law, and then
identifying the events of
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interest in calculating the
probability of that event.
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So now let's actually solve the
problem that we really are
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interested in, which is that
instead of confining Romeo and
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Juliet to arrive in only
15-minute minute increments,
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really, time is continuous,
and Romeo and Juliet can
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arrive at any time.
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So they don't necessarily have
to arrive 15 minutes late.
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Romeo could arrive 15 minutes
and 37 seconds late
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if he wanted to.
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So now our new sample space is
actually just, instead of only
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these 25 points in the grid,
it's this entire square.
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So any point within the square
could be a possible pair of
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meeting times between
Romeo and Juliet.
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So that is our new sample
space, our new omega.
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And now let's assign a
new probability law.
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And now, instead of being in the
discrete world, we're in
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the continuous world.
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And the analogy here
is to consider
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probabilities as areas.
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So the area of this entire
square is one.
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And that also corresponds to the
probability of omega, the
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sample space.
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And imagine just spreading
probability evenly across this
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square so that the probability
of any event--
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which in this case
would just be any
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shape within this square--
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is exactly equal to the
area of that shape.
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So now that is our new
sample space and our
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new probability law.
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So what we have to do now is
just to identify the event of
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interest, which is still the
event that Romeo and Juliet
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arrive within 15 minutes
of each other.
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So let's do that.
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If Romeo and Juliet arrive both
on time, then obviously
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they'll meet.
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And if Romeo's on time and
Juliet is 15 minutes late,
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then they will still meet.
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And in fact, any pairs of
meeting times between these
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would still work, because now
Romeo can be on time, and
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Juliet can arrive at any time
between 0 and 15 minutes late.
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But you notice that if Juliet is
even a tiny bit later than
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15 minutes, then they won't
end up meeting.
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So this segment here is part
of the event of interest.
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And similarly, this
segment here is
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also part of the event.
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And if you take this exercise
and extend it, you can
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actually verify that the event
of interest is this strip
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shape in the middle
of the square.
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Which, if you think about it,
makes sense, because you want
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the arrival times between Romeo
and Juliet to be close
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to each other, so you would be
expect it to be somewhere
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close to a diagonal
in this square.
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So now we have our event
of interest.
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We have our sample space and
our probability law.
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So all we have to do now is
just calculate what this
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probability is.
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And we've already said that
the probability in this
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probability law is just areas.
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So now it actually just boils
down to not a probability
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problem, but a problem
in geometry.
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So to calculate this area, you
can do it in lots of ways.
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One way is to calculate the area
of the square, which is
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1, and subtract the areas
of these two triangles.
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So let's do that.
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So in the continuous case, the
probability of meeting is
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going to be 1 minus the
area of this triangle.
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The base here is 3/4 and
3/4, so it's 1/2
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times 3/4 times 3/4.
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That's the area of one
of these triangles.
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There's two of them, so
we'll multiply by two.
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And we end up with 1
minus 9/16, or 7/16
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as our final answer.
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So in this problem, we've
reviewed some basic concepts
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of probability, and that's
also helped us solve this
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problem of helping Romeo and
Juliet meet up for a date.
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And if you wanted to, you could
even extend this problem
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even further and turn
it on its head.
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And instead of calculating given
that they arrive within
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15 minutes of each other, what
is the probability that
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they'll meet, let's say that
Romeo really wants to meet up
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with Juliet, and he wants to
assure himself a least, say, a
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90% chance of meeting Juliet.
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Then you can ask, if he wants to
have at least a 90% chance
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of meeting her, how long should
he be willing to wait?
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And so that's the flip
side of the problem.
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And you can see that with just
some basic concepts of
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probability, you can answer
some already pretty
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interesting problems.
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So I hope this problem was
interesting, and we'll
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see you next time.
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