# Thread: are they having an affair?

1. ## are they having an affair?

me and my friend at work have been having a debate .
there is a man and women at work that everyone suspects is having an affair.
one day out of the blue
they both phoned in sick on the same day , my friend said ,what are the odds of that happening ,
i started to wonder........
i said well if there are 240 working days in a year , and they both have exactly 1 day off a year

it would be 240 * 240 =57600 .. .. so 1 in 57,600
but if they both had exactly 3 days off a year it would be
240 / 3 = 80 ... 80 * 80 = 1 in 6,400
he disagrees

i know there are other things to consider like , we dont know exactly how many
days they both have off a year
but if the cases are as above ,the math is correct i think ?

2. 240 * 240 =57600 .. .. so 1 in 57,600
240 / 3 = 80 ... 80 * 80 = 1 in 6,400

3. yes im wondering to about that one!
im also wondering if there is a formula to work out these type of questions , for instance we dont no for sure
how many days people have off sick a year , maybe the average is , say 5... is there a formula to work it out from that...
my working out is scrappy at most ,

4. Hmm, I modified it a little bit to make it easier to look at.

Instead I'm looking at 5 possible days in which the 2 workers can sick at once.
This gave me 1 in 5.

I would maybe even say that this:
240 * 240 =57600 .. .. so 1 in 57,600
Might not be true, and the real number would be 1 in 240.

So the result for this:
240 / 3 = 80 ... 80 * 80 = 1 in 6,400
would actually be: 3 in 240 or 1 in 80

5. You need to study the Birthday Problem.
Scroll down to the probability discussion.
What is the probability the lady picks a different leave day?

6. no... i think your wrong ..... but i dont know for sure but i think that if there are 240 working days
and they both have exactly 1 day off a year .....and they pick the same day it is 240 *240 = 57,600

and if they both have exactly 3 days each off a year , it is 1 in 6,400
that they pick the same day

they are independent
the day he chooses to have sick, does not effect the day she chooses

7. Hello, jickjoker!

There is a man and women at work that everyone suspects is having an affair.
One day out of the blue, they both phoned in sick on the same day.
My friend said, "What are the odds of that happening"
i started to wonder . . .

But if they both had exactly 3 days off a year . . .

The man can have any 3 days-off of the 240 working days.

Consider the probability that the woman has none of the days-off that the man has.

There are: . $\displaystyle {240\choose3} \,=\,\frac{240!}{3!\,237!}$ possible choices.

To avoid the man's days-off, she must select 3 of the other 237 days.
. . There are: . $\displaystyle {237\choose3} \:=\:\frac{237!}{3!\,234!}$ ways.

$\displaystyle\text{Hence: }\:P(\text{no matches}) \;=\;\frac{{237\choose3}}{{240\choose3}} \;=\;\frac{237!}{3!\,234!}\cdot\frac{3!\,237!}{240 !} \;=\;\frac{219,\!067}{227,528}$

$\displaystyle\text{Therefore: }\:P(\text{some matches}) \;=\;1 - \frac{219,067}{227,528} \;=\;\frac{8461}{227,528}$

The probability that the man and woman have at least one day-off together

. . is: . $0.037186632$ . . . or about 1 in 27.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Stop arguing . . . Plato is absolutely correct!

If they each have one day-off per year,
. . the reasoning goes like this.

The man can have any day-off . . . It doesn't matter.

What is the probability that the woman has the same day-off?
. . Answer: . $\dfrac{1}{240}$

Your answer involving $240^2$ is a different problem.

What is the probability that the man and woman take June 23rd off?

$\text{We have: }\:\begin{Bmatrix}P(\text{Man, June 23}) &=& \frac{1}{240} \\ \\[-3mm] P(\text{Woman, June 23}) &=& \frac{1}{240} \end{Bmatrix}$

$\text{Then: }\:P(\text{Man and woman, June 23}) \:=\:\dfrac{1}{24)}\cdot\dfrac{1}{240} \:=\:\dfrac{1}{57,\!600}$

Get it?

8. Originally Posted by jickjoker
no... i think your wrong ..... but i dont know for sure but i think that if there are 240 working days
and they both have exactly 1 day off a year .....and they pick the same day it is 240 *240 = 57,600
Isn't that just the number of combinations and not the chance that they are sick on the same day.

Hmm maybe if we think like this:
Think about a number between 1 and 240, what are the chances of me guessing right? Well that would be 1 in 240 .. Let say I failed to guess the number.
Now think about a number between 1 and 240, what are the chances of me guessing right? Still 1 in 240.. It doesn't matter what number you pick or how many times I guess, my chances are still 1 in 240.

Same goes for Male Worker and Female Worker. It doesn't matter what day the Male Worker is sick, because the chances of the Female Worker also being sick is 1 in 240.

I might be right, I might be wrong.

9. mr Soroban
thank you for looking at this problem , i suspect that you are correct , i will look at your assesment tomorrow
i have drunk to much tonight ,but i thank you for your diagnosis ...doc