NWERC23-080.Date Picker

The NWERC is coming up and your agenda is filling up with meetings. One of your teammates wants to plan a meeting, and asks for your input. However, instead of asking you for your exact agenda, you have to fill out two separate polls: one for indicating which days you are available, and one for the hours!

As a computer scientist, you plan your meetings only on whole hours and each meeting takes an integer number of hours. Therefore, your agenda can be modelled as a matrix of $7$ rows (days), and $24$ columns (hours). Each cell in this matrix is either ‘.’ or ‘x’, meaning that hour of that day you are either free or have a meeting, respectively.

You have to pick at least $d$ days in the first poll and $h$ hours in the second poll, and we assume the meeting will take place on any of your picked hour/day combinations with equal probability. What is the probability that you can attend the meeting if you fill in the polls optimally?