Problem H
Cows and Cars
Input: standard input
Output: standard output
Time Limit: 4 seconds
In television
contests, participants are often asked to choose one from a set of or doors for
example, one or several of which lead to different prizes. In this problem we
will deal with a specific kind of such a contest. Suppose you are given the
following challenge by the contest presenter:
In front of you there are three doors. Two of them hide a cow, the other one
hides your prize - a car.
After you choose a door, but before you open it, I will give you an hint, by
opening one of the doors which hides a cow (I'll never open the door you have
chosen, even if it hides a cow). You will then be able to choose if you want to
keep your choice, or if you wish to change to the other unopened door. You will
win whatever is behind the door you open.
In this example, the probability you have of winning the car is 2/3 (as hard as
it is to believe), assuming you always switch your choice when the presenter
gives you the opportunity to do so (after he shows you a door with a cow). The
reason of this number (2/3) is this - if you had chosen one of the two cows,
you would surely switch to the car, since the presenter had shown you the other
cow. If you had chosen the car, you would switch to the remaining cow,
therefore losing the prize. Thus, in two out of three cases you would switch to
the car. The probability to win if you had chosen to stick with your initial
choice would obviously be only 1/3, but that isn't important for this problem.
In this problem, you are to calculate the probability you have of winning the
car, for a generalization of the problem above:
- The number of cows is variable
- The number of cars is variable (number of cows +
number of cars = total number of doors)
- The number of doors hiding cows that the
presenter opens for you is variable (several doors may still be open when you
are given the opportunity to change your choice)
You should assume that you always decide to switch your choice to any other of
the unopen doors after the presenter shows you some doors with cows behind it.
Input
There are several test cases for your program to process. Each test case
consists of three integers on a line, separated by whitespace. Each line has
the following format:
NCOWS NCARS NSHOW
Where NCOWS is the number of doors with cows, NCARS is the number of doors with
cars and NSHOW is the number of doors the presenter opens for you before you
choose to switch to another unopen door.
The limits for your program are:
1 <= NCOWS <= 10000
1 <= NCARS <= 10000
0 <= NSHOW < NCOWS
Output
For each of the
test cases, you are to output a line containing just one value - the
probability of winning the car assuming you switch to another unopen door,
displayed to 5 decimal places.
Sample input
2 1 1
5 3 2
2000 2700 900
0.66667
0.52500
0.71056