Jump or fall - what's the difference?  Either way the cat is seriously
injured or dead.  If you are going to open the windows, put sturdy
screens in them.

   Al the answer could be yes    If you ever watch America funniest home
videos  they show  hundred of cats falling of their perch when they are
asleep.
   Years ago in my other house we had a second story bay section window.
The furball who is now passed jumped up and hit the screen and it popped out
and he tumbled onto the balcony edge  luckily it was just a foot below  but
the furball mad one heck of a racket till I rescued him from his escapade.
Since than I use screens and metal adjustable window screens to prevent them
from knocking it out  and the furballs can still see out of them with no
problem but have no chance of knocking it out since it locks into the window
track and the window itself  holds it in place

Think of something like this to protect your furballs
http://www.drsfostersmith.com/Product/Prod_Display.cfm?pcatid=15736&Ntt=cat+wind
ow&Ntk=All&Ntx=mode+matchallpartial&Np=1&N=2002&Nty=1

Congrats on your decision to become a cat servent.   I have to second Ted's
reply...allowing a cat access to an unscreened open window is a recipe for
disaster - whether you're there to supervise or not.   If you're not up to
buying sturdy screens, a nice high cat tree by the window or a perch for them
to look out the closed window can be equally entertaining.  I put a birdhouse
outside the window where my cat tree is...its like an entertainment center
for them.  :>

YES. Keep the windows closed.
Gail

YES1    Get well fitted screens ASAP.  In the meantime keep windows shut.
MLB

As the others  have said--yes, you need screens, the cat will jump.

We lived in a second story apartment and ours went out the window one day
after a poorly directed "toy toss".  He was fine (quite startled--oh the
look on his face, sitting wide-eyed in the gravel driveway was CLASSIC) but
thereafter decided that was the perfect way to get out of the house and
wander when we refused to let him out. At first he'd push the storm window
open that was over the front porch, then crawl onto the roof and leap to the
adjacent tree and climb down.  After we learned to lock that window, he'd
simply go to any other open window and jump.

I saw a cat jump off a 3rd floor patio that was above mine,it was after
some ducks on the ground below.It was quite a leap.The owner never knew her
cat was missing,I had to knock on her door and when she opened the door,the
cat scooted in,like it was embarassed and afraid of the outside.

The simple answer is, YES.
They might not mean to jump, or fall....
But as life is, cats are very curious creatures with adventurous
natures, so the best thing to do is to assume that they will fall out,
or even simply investigating, & over stretching themselves. I know
that they say a cat has 9 lives, but that is fantasy in motion.
Fit some screens securely if you can, then there will be no worry @
all if they decide to investigate, what happens if they lean a bit too
far out of them...

Welcome to the cat slaves society & I look forward to meeting your new
addition to the family when you have the time to post a lovely photo
of your new kitty, please?
Many thanx,
Sheelagh