cs-101 — The Late Bus Frequency

Problem

From two weeks of bus logs, estimate the probability of a late arrival, and of two lates in a row (testing independence empirically).

Your deliverable is the count (or probability) with the sample space stated, the case analysis, and a brute-force check for a small instance. If the problem is ambiguous anywhere, state the assumption you are making and carry on — a stated assumption is part of a good answer.

Examples

• Input: 14 days, 4 late -> 4/14; two consecutive lates occurred once in 13 possible pairs
• Output: P(late) ~ 0.286; P(late twice in a row) ~ 1/13 ~ 0.077 vs P(late)^2 ~ 0.082 - close: near-independence

Solve for the worked instance first. Then ask what would change if one number, name, or condition were replaced by something stranger.

What to notice

• Estimate from frequencies, then TEST independence: does P(both) = P(late) x P(late)?
• 'How many total, how many favourable' is the whole skeleton of probability.
• Complements turn hairy unions into one subtraction.

Discussion starter

After the solution is revealed, the class spends five minutes on this prompt: "When is enumeration a legitimate answer rather than a crutch?"

Constraints and assumptions

• Difficulty: Beginner (no prior experience assumed).
• Topics: probability, frequencies.
• Classroom-scale inputs unless the problem says otherwise.
• Time-box: about five minutes for a first attempt; partial answers with clear reasoning are worth more than silent struggling.
• Optional stretch (not required): describe an alternative approach and say when it beats the obvious one.

Hints and extension

• First nudge: identify the pattern the worked instance uses before computing anything.
• If stuck after three minutes, write the input/output line you expect and trace the smallest example by hand.
• After the reveal, attempt the extension below before the next lecture.
• Extension challenge: Compute both conditionals from your table and explain the denominator swap.