[A simple counting problem].

In a given month, I have custody of my children on three weekends (6pm
fri--8am mon), and my wife has custody on the remaining days (ie,
weekdays and one weekend). There is a task that my wife and I must
perform together, The task takes two days; the two days can be any two
days of the month. We'll call a month four weeks, ie, 28 days.

My wife insists that the only two days on which she can perform the
task coincide with one of the weekends during which I am caring for my
children.

If all possible combinations of any two days in a given 'month' (four
weeks) are equally likely, what is the probability that my wife can
perform the task on only two consecutive days falling upon any one of
the three weekends on which I am caring for my children?

Hi yetanother!!


First of all we need to count all the combinations of any two days in
a given 'month':
We have 28 different days: Day 1, Day 2, ..., Day 28.
The number of the total combinations of any two days in a given 'month' is:

  C  = 28! / (26! * 2!) = 28*27 / 2 = 378
28 2 

Now we need to count the group of two consecutive days falling upon
any one of the three weekends:
Here I have a doubt about the statement of the problem, it is not
clear which are the days involved in the weekends on which you are
caring for your children for the purposes of the problem, it can be
four days per weekend (Fri, Sat, Sun and Mon - taking complete and
partial days together -) or two days per weekend (taking into account
only complete days: Sat and Sun).

In the first case (four days per weekend) there are 3 groups of two
consecutive days per weekend:
Fri-Sat; Sat-Sun; and Sun-Mon.
There are 3 weekends, then we have 9 groups of two consecutive days
falling upon days on which you  are caring your children.
This 9 groups of two days are part of the total number of groups of
any two days in a given 'month', and each group is equally likely, so
the probability that your wife can perform the task on only two
consecutive days falling upon any one of the three weekends on which
you are caring for your children is:
P = 9/378 = 0.0238 = 2.38%


In the second case (two days per weekend) there are 1 group of two
consecutive days per weekend.  There are 3 weekends, then we have 3
groups of two consecutive days falling upon days on which you  are
caring your children.
This 3 groups of two days are part of the total number of groups of
any two days in a given 'month', and each group is equally likely, so
the probability that your wife can perform the task on only two
consecutive days falling upon any one of the three weekends on which
you are caring for your children is:
P = 3/378 = 0.008 = 0.80%

----------------------------------------------------------

For additional references see:

"PERMUTATIONS AND COMBINATIONS":
http://www.themathpage.com/aPreCalc/permutations-combinations.htm

"College Algebra Tutorial on Combinations":
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut57_comb.htm


I hope that this helps you. Feel free to request for further
assistance if you feel that you need it or if you find something wrong
or unclear.


Best regards.
livioflores-ga

A quick note in possible defense of your wife:

Weekends are typically days that people have free.  If she works most
of the week or if your children are in school then the only possible
time she would have is on the weekend.  And if she has the kids only 1
weekend of the month then it is also likely that she has that weekend
very planned of things she wants to do with them.
If weekdays are in fact likely candidates for when this activity could
logistically be done then the probabilities that livioflores worked
out are very much correct, but do be careful that you take everything
into consideration.  I'd also suggest that if your wife gets partial
custody for a weekend then you have good grounds to request extra time
(maybe on the evenings) with your kids this month.

Good luck!  I know custody issues are never an easy thing to make
happen for either the parents or the kids.

Thanks, all. I had assumed Sat/Sun, but perhaps that was hasty. 
Anyway, yes, these are difficult issues. IMHO I've been the one to
give on just about everything that's come up, but somehow she can do
this activity only my custody days.  Her lawyer insists that I be
present and that the kids not be present, and so for this month she'd
have the children 22 days, 4 days for me, and 2 for the babysitter. 
And this is at the end of a temporary (4 month) custody arrangement
that gave each of us 50% custody. Yes, difficult times.