Uniform Probabilities on a Square

What this lecture covers

This recitation problem asks: if two people each arrive at a lunch date at a random time within an hour of noon, and each waits only 15 minutes for the other, what is the probability they actually meet? It's used to practice building a sample space and probability law from scratch, and to connect a discrete version of a problem to its continuous counterpart.

The video first solves a simplified, discrete version of the problem (arrivals only in 15-minute increments) before extending it to the real, continuous case, where probability is measured as area.

Key ideas

• Grid sample space: pairing each person's arrival time as a point on a two-dimensional grid or square makes the geometry of the problem visible.
• Discrete uniform law: when every grid point is equally likely, probability is just a count of favorable outcomes divided by the total.
• From discrete to continuous: refining the arrival-time increments turns the grid into a continuous square, and the probability law becomes area instead of a point count.
• Probability as area: for a uniform continuous model, the probability of an event is the area of its region divided by the area of the whole sample space.
• Identifying the event geometrically: "arrive within 15 minutes of each other" becomes a diagonal strip on the square, and its area is found by subtracting two corner triangles from the full square.

Before you watch

• Know the discrete uniform law and the idea of assigning probability as area, both introduced in the first lecture of this course.
• Basic geometry (area of a square and a triangle) is needed for the final calculation.

Check your understanding

1. Why does restricting arrival times to 15-minute increments make the problem easier to solve first?
2. How does the sample space change when arrival times become continuous instead of discrete?
3. Why does the region "they meet" form a diagonal strip on the square, and how is its area computed?
4. If each person were willing to wait 30 minutes instead of 15, how would the strip's area change?