Trying to figure out the probability of sick days for an employee who I think is lying We have recorded $7$ individual sick days with $4$ of the sick days falling on a Monday which is a little suspicious. Let's assume that there are no public holidays and it is just a standard working week, Monday to Friday.
What is the probability/chance of this? 
It would be helpful to understand how to work this out!
 A: tl;dr: You might spend less time wondering about peoples' sick days. :-)
It depends a lot on what you consider suspicious, and also (possibly) how many employees you're keeping tabs on.
In the most restrictive case, you only consider Monday suspicious, and are only keeping tabs on this one employee.  In that case, the probability (given the null hypothesis that the employee is equally likely to be sick on any given day of the week) that at least four of seven sick days fall on a Monday is
\begin{align}
P & = \binom{7}{4} \left(\frac{1}{5}\right)^4 \left(\frac{4}{5}\right)^3 \\
  & + \binom{7}{5} \left(\frac{1}{5}\right)^5 \left(\frac{4}{5}\right)^2 \\
  & + \binom{7}{6} \left(\frac{1}{5}\right)^6 \left(\frac{4}{5}\right)^1 \\
  & + \binom{7}{7} \left(\frac{1}{5}\right)^7 \left(\frac{4}{5}\right)^0 \\
  & = \frac{2240+336+28+1}{78125} \\
  & = \frac{2605}{78125} = \frac{521}{15625} \doteq 0.033344
\end{align}
or about $1$ chance in $30$.  Just to be clear, that is not the probability that the employee is malingering.  It is the probability, given that the employee is not malingering, that they would exhibit this sort of sick-day record.  This level of result is generally not considered statistically significant.
And note that this is given that you're only looking at one employee.  If you work in an office with ten employees, say, it's far more likely that you'll notice one employee with this kind of record, purely by dumb luck, even if none of them is malingering.  The probability there is given by
$$
P = 1 - \left(1-\frac{521}{15625}\right)^{10} \doteq 0.28761
$$
or over a $1$ in $4$ chance that at least one of the employees will be out at least four Mondays out of seven sick days.  It's really not that unusual.
And furthermore, this assumes that only Mondays are suspicious.  Why shouldn't Fridays be suspicious, too?  That raises the odds of a completely innocent trend correspondingly.
