Commuter Survey and Conditional Probability
A transportation office surveys 400 commuters about their primary mode of travel and whether they arrive late more than twice a week. Results: 160 drive alone, of whom 40 are frequently late; 120 take the bus, of whom 30 are frequently late; 80 carpool, of whom 10 are frequently late; and 40 bike or walk, of whom 0 are frequently late. The office wants to use these numbers to argue for expanded bus service, but a skeptical analyst points out that raw counts can mislead and asks for probabilities computed carefully.
1. Compute the probability that a randomly selected commuter is frequently late, and compute the probability that a commuter is frequently late given that they take the bus. Explain why these two numbers differ.
Read the worked answer
Total frequently late = 40 + 30 + 10 + 0 = 80. So P(late) = 80/400 = 0.20.
Bus riders = 120, of whom 30 are late. So P(late | bus) = 30/120 = 0.25.
The conditional probability is higher because the base group changed: we restricted the sample space to bus riders only. P(late) uses all 400 commuters as the denominator, while P(late | bus) uses only the 120 bus riders.
2. The analyst claims bus riders are 'the latest group.' Evaluate this claim by comparing the late-rate across all four modes, and state whether the claim is supported.
Read the worked answer
Late rate by mode:
- Drive alone: 40/160 = 0.25
- Bus: 30/120 = 0.25
- Carpool: 10/80 = 0.125
- Bike/walk: 0/40 = 0.00
Bus riders (0.25) tie with drive-alone commuters (0.25), so the claim that bus riders are 'the latest group' is not supported - they are no later than drivers. The raw count of 30 late bus riders is smaller than 40 late drivers, but that reflects group size. Comparing rates, not counts, is the correct comparison.