The length of time patients must wait to see a doctor at an emergency 
room in a large hospital is uniformly distributed between 40 minutes 
and 3 hours.

A uniform distribution is rectangular shaped and has area of 1 
Its base extends from 40 minutes to 180 minutes (which is the 
same as 3 hours). The area of 1 represents the fact that a 
patient will have to wait somewhere between 40 minutes and 180 
minutes is certain, and has probability of 1 which is equal to 
that area of 1.

      ______________________________________________________
     |                                                      |
     |                                                      |     
    40  50  60  70  80  90 100 110 120 130 140 150 160 170 180 

The base of that rectangle is 180-40 or 140 units long.  The 
area of the entire rectangle is 1, using A = base×height, we 
have that the rectangle is 1/140th of a unit high.

a) What is the probability that a patient would have to wait 
between 50 minutes and two hours? 

Since two hours is 120 minutes, that probability is represented 
by the area of the rectangle whose base extends from 50 to 120, 
i.e., the one containing the #'s below:

      ______________________________________________________
     |  |###########################|                       |
     |  |###########################|                       |     
    40  50  60  70  80  90 100 110 120 130 140 150 160 170 180 

Its base is 120-50 or 70 units wide and its height is 1/140,
so the probability is 70×1/140 or 70/140 or 1/2

b) What is the probability that a patient would have to wait exactly 
one hour? 

I'm going to come back to this one after I have answered all the
others because it's different from the others.

c) What is the probability that a patient would have to wait between 
60 and 90 minutes? 

That probability is represented by the area of the rectangle whose 
base extends from 60 to 90, i.e., the one containing the #'s below:

      ______________________________________________________
     |       |###########|                                  |
     |       |###########|                                  |     
    40  50  60  70  80  90 100 110 120 130 140 150 160 170 180 

Its base is 90-60 or 30 units wide and its height is 1/140,
so the probability is 30×1/140 or 30/140 or 3/14


d) What is the probability that a patient would have to wait between 
120 minutes and 3 hours? 


Since three hours is 180 minutes, that probability is represented 
by the area of the rectangle whose base extends from 120 to 180, 
i.e., the one containing the #'s below:

      ______________________________________________________
     |                              |#######################|
     |                              |#######################|      
    40  50  60  70  80  90 100 110 120 130 140 150 160 170 180

Its base is 180-120 or 60 units wide and its height is 1/140,
so the probability is 60×1/140 or 60/140 or 3/7
    

e) What is the probability that a patient would have to wait between 
60 and 120 minutes?

That probability is represented by the area of the rectangle whose
base extends from 60 to 120, i.e., the one containing the #'s below:

      ______________________________________________________
     |       |######################|                       |
     |       |######################|                       |     
    40  50  60  70  80  90 100 110 120 130 140 150 160 170 180

Its base is 120-60 or 60 units wide and its height is 1/140,
so the probability is 60×1/140 or 60/140 or 3/7


Now for the (b) part

b) What is the probability that a patient would have to wait exactly 
one hour? 

This one is different!  It is represented by the area of the 
"rectangle" whose base starts at 120 AND ENDS at 120.  Wow! What kind
of "rectangle" is that???  It is a "rectangle" which is so thin that it
is only a line segment drawn at 120 (since 1 hour = 120 minutes). 


      ______________________________________________________
     |                              |                       |
     |                              |                       |     
    40  50  60  70  80  90 100 110 120 130 140 150 160 170 180


So its base is ZERO, so its area is ZERO!!! That means the probability
that someone will wait EXACTLY 120 minutes is ZERO, which means that it
is IMPOSSIBLE for someone to wait EXACTLY one hour!! That seems so 
wrong!! But remember, we are looking at it from a practical standpoint.
It is IMPOSSIBLE to measure anything PERFECTLY!  No clock or timing 
device can measure any waiting time of EXACTLY one hour PERFECTLY!  It 
is always within at least a fraction of a minute or second, depending 
on how accurate our time-measuring device is calibrated.  So we must 
consider it IMPOSSIBLE that a patient waits EXACTLY one hour, simply 
because it is IMPOSSIBLE to measure EXACTLY one hour PERFECT PRECISELY.  
That may seem strange at first, but if you think about it -- nothing's
perfect!!!!! So the answer is 0.

Edwin