The city Metro train on the orange line
runs every 15 minutes during the time
between 6:00 PM and 9:00 PM.
Let X be the waiting time of a randomly selected passenger
between 6:00 PM and 9:00 PM.
The waiting time can be anytime from zero minutes
to 15 minutes with an equally likelihood of occurring.
And therefore we can say the continuous random variable X
has a uniform distribution with a minimum
of zero minutes and a maximum of 15 minutes.
We indicate this using the notation shown here below
where A is the minimum and B is the maximum.
So we got the random variable X has a uniform distribution
with a minimum of zero and a maximum of 15.
Number two, we're asked to use random variable notation
to express symbolically each of the following.
First, an event in which the waiting time
of a randomly selected passenger is less than 12
or less than 12 minutes.
This would be X less than 12.
Next, the probability of an event in which the waiting time
of a randomly selected passenger is less than five.
This is the probability that X is less than five.
And then finally we have the probability
of an event in which the waiting time
of a randomly selected passenger is less than eight
is equal to 0.53.
This is the probability
that X is less than eight equals 0.53.
And now let's find some probabilities.
We want to find the probability
that it takes less than 12 minutes to wait for the train,
more than five minutes to wait for the train
and between four and 12.5 minutes to wait for the train.
We'll find these by hand
and then verify the results using desmos.com.
This first set up the graph of the distribution shown below
because we have a uniform distribution.
We know the probably density function
is a horizontal line graft here.
The minimum is zero, the maximum is 15
And they're probability density function is,
F of X equals one divided by the quantity B minus A,
where a is equal to zero and B is equal to 15.
Which gives us F of X equals one divided by the quantity 15
minus zero or just 1/15.
So now we know that our probability density function
is F of X equals 1/15,
which means any rectangle formed using the X axis
and the probability density function
has the height of at 1/15.
This length here.
That should make sense because notice how the total width
of the interval from zero to 15 is 15 units.
And we know the total area must be equal
to one or 100% and 15 times 1/15 is one.
So let's first find the probability it takes less 12 minutes
to wait for the train
which is the probability that X is less than 12.
Let's say 12 is here.
The probability that X is less than 12
is equal to the area under the probability density function
from zero to 12, this area here.
So let's go ahead and find that area.
Let's use the formula,
the area is equal to the base times the height.
Well, the base is equal to 12 minus zero.
It is 12 times the height, which is always 1/15.
12 times 1/15 is 12/15.
And 12/15 simplifies the 4/5.
We are told to express the answer as a decimal
for fx is equal to 0.8.
Next you want the probability it takes
more than five minutes to wait for the train,
which is the probability that X is greater than five.
So let's say five is here.
The probability that X is greater than five
is equal to the area under the probability density function
from five to 15, this area here.
The area again, is equal to the base times the height.
The base is equal to 15 minus five times the height of 1/15.
This gives us 10 times 1/15 or 10/15
which simplifies the 2/3.
2/3 is equal to 0.6 repeating.
We're told to round to two decimal places.
And therefore we have approximately 0.67.
And then for the last part
we want the probability
that X is between four and 12.5 minutes
which is X greater than four and less than 12.5.
So let's say four is here.
12.5 is approximately here.
The probability the waiting times
are between four and 12.5 minutes
is equal to the area under there probably density function
from four to 12.5.
This gives us the area is equal to the base,
which is 12.5 minus four times the height of 1/15.
12.5 minus four is 8.5.
8.5 times 1/15 is 8.5/15.
And let's go ahead and go to the calculator
and convert to a decimal.
Let's do this on Desmos.
So going to Desmos quick graphing calculator
enter 8.5 divided by 15 enter.
Running to two decimal places we have approximately 0.57.
Let's verify these probabilities using Desmos.
Going back to Desmos in cell two let's click functions
click the distributions tab, click uniform distribution
enter the mean of zero, max 15.
Close the Desmos key pad, click zoom fit notice.
Notice now we have a nice graph
of the uniform distribution on the right.
To find the probability that X is less than 12,
click the find the cumulative probability box.
For the minimum value, enter zero,
for the maximum value enter 12.
And notice how it does give us a probability of 0.8,
which we found by hand.
To find the probability that X is greater than five,
we're going to change the minimum to five
and change the maximum to 15
and notice how we do get 0.6 repeating,
which is approximately 0.67.
Notice how it also shows us the area on the right.
And for the last probability,
where the probably that X is greater than four
and less than 12.5, the minimum is now four
and the max is now 12.5.
And that is how we do get approximately 0.57
which also shows as the area on the right.
I hope you found this helpful.