The Odds of Three People Sitting Next to Each Other on a Plane

The probability of three specific individuals sitting next to each other on a plane is an intriguing question that can be calculated mathematically. Let's explore the factors and the methodology behind determining this chance.

Factors Influencing the Probability

The probability of three individuals sitting next to each other on a plane depends on several factors, including the total number of passengers and the seating arrangements.

Simplified Calculation

Assumptions

:n - We assume the plane has a standard seating configuration, such as 3 seats per row. :n - There are a total of n passengers on the flight, including the 3 specific individuals.

Total Ways to Arrange Passengers

The total number of ways to arrange n passengers is given by n!.

Ways for 3 People to Sit Together

To calculate the number of ways 3 specific people can sit next to each other, we treat them as a single unit. This simplifies the problem to arranging n - 2 units.

The number of ways to arrange the block of 3 and the other passengers is (n - 2)!. Within the block, the 3 people can be arranged in 3! ways.

Total Arrangements with the Block

The total number of arrangements with the block is (n - 2)! * 3!.

Probability

The probability of the 3 people sitting next to each other is given by:

P{3 together} (n - 2)! * 3! / n! 6 / (n * (n - 1) * (n - 2))

Example Calculation

Suppose there are 150 passengers on the plane, including the 3 specific individuals:

P{3 together} 6 / (150 * 149 * 148) ≈ 0.00000025

General Strategy with Specific Plane Configuration

For a plane with 120 seats, we can use the combination formula to calculate the total number of ways to choose 3 non-repeating seats:

n! / (r! * (n - r)!) where n 120 and r 3.

This gives us 280,840 combinations.

Since a typical 737-800 plane has 20 rows of 6 seats with 3 on a side, there are 40 possible groups of 3 where you could sit. Thus, the probability of sitting together is:

40 / 280,840 1 / 7,021

Conclusion

The odds of three specific people sitting next to each other are dependent on the total number of passengers. As the number of passengers increases, the probability decreases significantly.

Understanding the underlying mathematics and probability can help us appreciate the sometimes surprising odds of such specific events occurring on a crowded airplane.