# Chances Are

I just came from a great weekend escapade, with my wife and two colleagues, in one of the beautiful regions of the Philippines. Bicol is home to Mayon Volcano, one of the seven perfect coned volcanoes on Earth. The scenery is so green and the air so refreshing. It was one of the best trips I ever made in the Philippines.

The trip was also full of coincidences. We met a colleague in one of the restaurants in Naga City where we had our dinner. In the same restaurant we also met one of our contacts in Ateneo de Naga. On the way home, we met another colleague at the airport. But the first coincidence happened on our way to Legazpi City where one of my companions (i’ll hide the name to protect the innocent) met a good friend in the airplane seated next to him. It was a pleasant surprise for my companion and he asked “What are the odds of that happening”? That’s a good question which got me thinking later that night.

To simplify the problem, let us not compute the odds of my companion and his friend to be in the same plane going to Bicol. Rather, let us compute the odds of him being seated next to his friend. The plane was an airbus A320 and about 180 seats. The seats are laid out in such a way that there is a middle isle and each row is composed of three seats. Let us draw a row of seats and assume we have persons A and B. How many ways can they be seated in that row in such a way that they sit next to each other? Below is a diagram on the ways two persons can be seated next to each other.

As you can see, there are four ways for them to be seated next to each other for any given row. In the first arrangement, we have A, B seated in that order. The third seat can be occupied by anyone. Having chosen the seats for A and B, how many choices do we have for the third person? Since there are 180 seats minus two, the third person has 178 other seats to choose from. Once person number 3 chooses one seat, the fourth person can then choose from the remaining 177 seats. Doing this for all other passengers, you will come up with 178! ways of assigning seating arrangements for the 180 passengers such that persons A and B are seated next to each other. Now, that is just for the case where A and B are in the first two seats. For the other three cases, you will see that it also takes 178! ways for each case. Since there are 4 cases per row and 60 rows, we have

$\displaystyle 4\cdot 60\cdot 178!$

ways to have two persons sit next to each other on a 180 seater plane. Since the total number of ways to assign passengers to 180 seats is 180!, then the probability of having two persons seat next to each other is

$\displaystyle \frac{4\cdot 60\cdot 178!}{180!}$
$\displaystyle = \frac{4\cdot 60\cdot 178!}{180\cdot 179\cdot 178!}$
$\displaystyle = \frac{4\cdot 60}{180\cdot 179}$
$\displaystyle = 0.00744879$

This is quite a low probability and it assumes that the seating arrangement is totally random. By totally random, I mean that my wife and I could be seated far apart from each other. In reality, this is not the case since many passengers come in groups and they prefer to be seated next to each other. If we take this into account, then the probability of my companion and his friend to be seated next to each other increases.