If 2 people have 2 days off per week each, what are the odds that they would have at least one day off that is the same?
How do you solve this? Any help?

  • 1
    $\begingroup$ What is the probability that all are different? $\endgroup$ – saulspatz May 9 '18 at 20:00
  • $\begingroup$ The probability of getting any combination of off days off is the same for the person. Ex.: (Monday/Tuesday's probability is the same as Tuesday/Sunday) $\endgroup$ – Filipe F. May 9 '18 at 20:04
  • $\begingroup$ For each person, are the two days off consecutive? $\endgroup$ – Joffan May 9 '18 at 20:05
  • $\begingroup$ No, they can be any combination of days $\endgroup$ – Filipe F. May 9 '18 at 20:11
  • 1
    $\begingroup$ Welcome to MSE! As a minimum you're expected to show what you have tried and where you are stuck. $\endgroup$ – samerivertwice May 9 '18 at 20:16

There are $\binom{7}{2}^2$ ways they can choose their days off. Of these, $\binom{7}{2}\binom{5}{2}$ involve no overlap. Therefore, the probability of some overlap is $1-\dfrac{\binom{5}{2}}{\binom{7}{2}}=1-\dfrac{5\times 4}{7\times 6}=\dfrac{11}{21}$.

  • $\begingroup$ @Joffan I advise you to make that a separate question. That'll give you room to decide what counts as adjacent. For example, are the first & last days of the week adjacent? $\endgroup$ – J.G. May 9 '18 at 20:05
  • $\begingroup$ I arrived to the same answer, but I was asking myself how to properly explain to my peers. I couldn't properly explain why you would use the (5|2) part? (sorry I can't format it properly) $\endgroup$ – Filipe F. May 9 '18 at 20:14
  • $\begingroup$ @FilipeF. Because if you try to list the no-overlap combinations, each time you specify one person's days off only $5$ days are available for the second person to choose from. $\endgroup$ – J.G. May 9 '18 at 20:18
  • $\begingroup$ @FilipeF. Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig May 10 '18 at 10:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.