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Probability · Lecture 18 of 76 · 11:55

Sampling People on Buses

Sampling People on Buses on YouTube

Study guide

What this lecture covers

This recitation problem uses the classic "inspection paradox" setup, four buses carrying different numbers of students, to practice writing PMFs and computing expected values while also building intuition. It compares two random variables: the size of the bus of a randomly chosen student, versus the size of the bus of a randomly chosen driver.

After watching, you should be able to write out a PMF for a random variable defined by sampling proportional to group size versus sampling uniformly over groups, compute expected values from those PMFs, and use an extreme-case argument to predict which of two expectations should be larger before doing the calculation.

Key ideas

  • Sampling a student vs. sampling a bus: choosing a random student weights larger buses more heavily, while choosing a random bus driver treats every bus equally regardless of size.
  • Building intuition with extreme cases: pushing the numbers to an extreme (one bus with 1 student, another with 1,000) makes it clear that sampling students is biased toward the large bus, while sampling drivers gives each bus a 50/50 chance.
  • PMF of X (student-sampled bus size): p_X(x) = (number of students in that bus) / 148 for each of the four bus sizes.
  • PMF of Y (driver-sampled bus size): p_Y(y) = 1/4 for each of the four bus sizes, since each driver is equally likely to be picked.
  • Result: E[X] (about 39) is larger than E[Y] (37), confirming that sampling people rather than groups biases the outcome toward larger groups.

Walkthrough

Setting up the two random variables (1:00)

Four buses carry 40, 33, 25, and 50 students (148 total). X is the size of the bus containing a uniformly random student; Y is the size of the bus of a uniformly random driver. The question asks which has the higher expected value.

Building intuition with an extreme case (3:00)

Before calculating, the lecture considers two buses of very different sizes (1 and 1,000 students) to argue that sampling a student almost always lands on the large bus, while sampling a driver is a coin flip between the two buses, predicting E[X] > E[Y].

Writing the PMFs and computing the expectations (5:03)

The PMFs for X and Y are written out explicitly and plotted, showing X's PMF favors larger bus sizes while Y's PMF is uniform across the four buses. Computing the weighted sums gives E[X] ~ 39 and E[Y] = 37, confirming the intuition from the extreme case.

Before you watch

  • Review probability mass functions and expected value from earlier lectures in this course, since this problem is a direct application of both.

Check your understanding

  1. Why does sampling a random student bias the outcome toward larger buses, while sampling a random driver does not?
  2. How does considering an extreme case (very unequal bus sizes) help predict the answer before doing the calculation?
  3. Why do X and Y share the same possible values but have different probabilities assigned to those values?

Vocabulary

inspection paradox (noun)
The surprising fact that sampling individuals from groups can favor larger groups, biasing the average.
The inspection paradox explains why the student-sampled bus looks bigger on average.
bias (statistical) (noun)
A systematic tendency for a measurement to lean toward one direction.
Sampling students creates a bias toward larger buses.
extreme case (noun)
An exaggerated example used to test or build intuition about a general rule.
Considering an extreme case makes the expected bias obvious.
weighted sum (noun)
A total formed by multiplying each value by its own probability before adding.
Expected value is a weighted sum over all possible bus sizes.
classic (setup) (adjective)
Widely known and often used as a standard teaching example.
This is the classic inspection paradox setup.
proportional (to size) (adjective)
Increasing or decreasing at the same rate as another quantity's size.
Sampling proportional to group size favors larger groups.
uniformly (at random) (adverb)
With every option given exactly the same chance.
A driver is picked uniformly at random among the four.
plot (a PMF) (verb)
To draw a graph showing a distribution's values.
The PMFs of X and Y are plotted side by side.
coin flip (noun)
A random event with two equally likely outcomes, used as a comparison.
Sampling a driver is like a coin flip between the two buses.
predict (verb)
To state in advance what a result is likely to be.
The extreme case is used to predict which expectation is larger.
confirm (intuition) (verb)
To show that an initial guess turns out to be correct.
The calculation confirms the intuition from the extreme case.
student (sampled) (noun)
An individual chosen at random from a group for a probability example.
A randomly sampled student defines the random variable X.
driver (noun)
The person operating one of several groups, used here to sample by group.
Sampling a random driver treats every group equally.
grouped by (phrase)
Organized according to a shared category or property.
Buses are grouped by how many students ride each one.
compare (two variables) (verb)
To examine two quantities side by side to see which is larger.
The problem compares two random variables built from the same buses.
sizable difference (noun)
A large enough gap between two values to matter.
There is a sizable difference between the two expected values.
classic setup (noun)
A well-known problem structure often used to teach a concept.
This uses the classic inspection paradox setup with buses.
total (students) (noun)
The overall sum across all groups combined.
The four buses carry a total of 148 students.
question (posed) (noun)
The specific thing being asked in a problem.
The question asks which of the two expectations is larger.
practice application (noun)
An exercise that applies a known concept to a new situation.
This is a practice application of PMFs and expected value.

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
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