Introduction

When multiple cars are parked in a rectangular parking space and each car has the possibility of moving in either direction, the likelihood of a collision between two or more cars becomes a relevant concern. This scenario can be analyzed through the principles of probability, which can help us understand the risks involved and perhaps guide us towards safer parking practices.

Scenario Explanation

Imagine a rectangular parking space where cars are parked on each of the four corners. Each car has the ability to move in either of its two directions, typically towards the center of the parking space. For the purpose of this analysis, we will assume the roads are not wide enough to allow passing, significantly limiting the chances of avoiding a collision.

Assumptions

The cars do not move diagonally. Each car is restricted to moving along the edges of the parking space. The roads are not wide enough for two cars to pass each other. This assumption is crucial, as it forces the cars to either stop or potentially collide with each other. The drivers are not perfect and might not be observant enough. This introduces a slight residual risk of collision beyond the strict no-passing rule.

Calculating the Probability

To determine the probability of a collision, we need to consider the possible movements of each car and then calculate the scenarios where at least two cars collide.

Each car has a binary choice: move left or right. Therefore, for four cars, the total number of possible movement combinations is 2416.

Probability of No Collisions

The only scenario where no collisions occur is when all four cars are moving in the same direction. This can happen in two ways: all cars moving left or all cars moving right.

The probability of all cars moving left is 0.540.0625. Similarly, the probability of all cars moving right is also 0.540.0625.

Therefore, the total probability of no collisions is 0.0625 0.06250.125.

Probability of Collisions

The probability of at least one collision occurring is the complement of the probability of no collisions. Thus, the probability of at least one collision is:

1-0.1250.875.

This indicates that in 87.5% of the cases, there will be at least one collision involving two or more cars.

Conclusion

The high probability of collisions in this parking space setup highlights the importance of clear signage and strict traffic rules to ensure the safety of all drivers. It also emphasizes the need for drivers to maintain situational awareness and follow traffic rules to minimize the risk of accidents.

References

[1] Traffic Safety and Signage

[2] Management of Rectangular Parking Spaces