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Probability · Lecture 4 of 76 · 9:16

Uniform Probabilities on a Square

Uniform Probabilities on a Square on YouTube

Study guide

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?

Vocabulary

grid (noun)
A pattern of evenly spaced points or lines used to organize possibilities.
Pairing two arrival times gives a point on a grid.
increment (noun)
A small, fixed step used to divide a range into pieces.
Arrival times are first limited to 15-minute increments.
refine (verb)
To make something more precise by using smaller steps or more detail.
Refining the time increments turns the grid into a continuous square.
diagonal (adjective)
Running at a slant, from one corner toward the opposite side.
The event of meeting forms a diagonal strip on the square.
strip (noun)
A narrow band-shaped region within a larger area.
The strip's area represents the probability that they meet.
corner triangle (noun)
A triangular region cut off at the corner of a larger shape.
The strip's area is found by subtracting two corner triangles.
arrival time (noun)
The moment at which someone or something reaches a place.
Each person's arrival time is picked at random within the hour.
simplified version (noun)
An easier form of a problem used to build understanding before the full version.
The simplified version only allows arrivals every 15 minutes.
counterpart (noun)
A matching version of something in a different setting.
The continuous case is the real counterpart of the discrete example.
geometry (noun)
The branch of math dealing with shapes, sizes, and positions.
The geometry of the square makes the event easy to visualize.
meet (each other) (verb)
To be at the same place at close enough times to encounter one another.
The two people meet if their arrival times are within 15 minutes.
wait (for someone) (verb)
To stay in place until another person arrives.
Each person is willing to wait 15 minutes for the other.
identify (an event) (verb)
To recognize and describe exactly which outcomes belong to an event.
The lecture identifies the meeting event geometrically on the square.
practice (verb)
To repeat an exercise in order to build a skill.
This problem is used to practice building a sample space from scratch.
classic (adjective)
Well known and often used as a standard example.
This is the classic Romeo and Juliet meeting problem.
coordinate (noun)
A number that gives a position along one axis of a graph.
Each person's arrival time becomes one coordinate on the square.
restrict (verb)
To limit something to a smaller range of possibilities.
Restricting arrival times to 15-minute steps simplifies the problem.
geometrically (adverb)
In a way based on shapes and positions rather than pure numbers.
The event is described geometrically as a diagonal strip.
willing (adjective)
Ready and prepared to do something.
Each person is willing to wait only a short time.
practice building (phrase)
Working through an exercise to gain skill in constructing something from scratch.
This problem is good practice building a sample space and probability law.

Chapters

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Jimmy Li

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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