Sampling People on Buses

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?