Math Is Fun Forum

4 couples are going on vacation to a 5-star hotel in Martinique. The receptionist gave them four consecutive rooms with numbers from 1 to 4. The women went upstairs to choose rooms (and each of them stayed to her own room), while their husbands were having a drink at the bar. After some drinks, they decided to go upstairs to their rooms. 
"If we all go upstairs and start opening the 4 doors at random until we find our wife, what are the chances we get to the correct room with maximum 2 attempts each?" the first man asked.

"This is easy", says the second. "Since we will open 2 out of 4 doors, each of us has 50% probability to locate his wife. So, in order for all four of us to make it, the probability is (1/2)^4, that is, 1/16."

"Wrong!" says the third. There is a way to get to the correct room with probability greater than 40% and without havng to communicate with our wives or between us, after the first of us goes upstairs."

What is the method he is suggesting?

We have a grid of dots m x n, where m is the number of rows and n the number of columns, where m≠n and m, n >2.
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Two players alternate turns and choose one dot from the grid and then pick all dots above and to the right of these dots. The purpose of the game is to force your opponent take the last dot.
Do I play first or second? What is the strategy to win?
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(for example I chose the 4th dot of the 3rd row from the bottom, so I got all 16 dots above and to the right of this one).

A public bus arrives at the bus stop sometime between 6:00 and 6:30.
A passenger arrives at the same bus stop some random time within this time interval. What is the probability that the passenger catches the bus within 5 minutes from the time he arrives at the stop?