Alex, Bob, and five friends are randomly assigned seats on a train. Six of them will sit in two rows of 3 that face each other, while the last person will have to stand. What is the probability that Alex and Bob are sitting directly across from each other?
The probability that Alex isn't standing is 6/7.. If Alex isn't standing, there is exactly one seat across from him and 6 positions left (the 5 seats and 1 standing), so the probability that Bob gets that seat given that Alex is seated is 1/6.
Thus, the overall probability is 6/7⋅1/6=1/7.
