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