Bus to Subway: Simulate the Connection

Model delays, run trials, and use the results to plan trips.

Design the Model

A trip uses one city bus followed by one subway train. The bus is late on 30% of trips, and the subway is late on 20% of trips. Assume the two delays are independent. Random digits 0–9 are equally likely.

Which assignment correctly models whether the bus is late?

A. Use 0–2 for late and 3–9 for on time.B. Use 0–1 for late and 2–9 for on time.C. Use 0–6 for late and 7–9 for on time.D. Use 0–3 for late and 4–9 for on time.

Which procedure correctly models one complete trip and preserves independence?

A. Generate one digit and use it for both the bus and the subway.B. Generate two independent digits: one for the bus and one for the subway.C. Generate a bus digit, then choose a different digit for the subway.D. Generate digits until one vehicle is late, then end the trip.

For the rest of the worksheet, use two independent digits per trip. The first digit models the bus: 0–2 means late. The second models the subway: 0–1 means late. All other digits mean on time.

The planner wants to estimate the probability that both vehicles are late. Which pair counts as a success? Read the bus digit first.

A. The pair (4, 8) counts as a success.B. The pair (1, 0) counts as a success.C. The pair (2, 6) counts as a success.D. The pair (7, 1) counts as a success.

A student plans to simulate 75 complete bus-and-subway trips using this model.

How many random digits are needed? digits

Run and Record

Run the simulation using the supplied random digits. Treat each row as one complete trip, even when a digit repeats.

Ten Simulated Trips
Trial	Bus digit	Subway digit
1	0	1
2	7	0
3	2	8
4	6	4
5	1	0
6	9	7
7	4	1
8	2	1
9	0	6
10	8	3

Select every trial in which both vehicles are late.

Use all ten trials to estimate the probability that exactly one vehicle is late.

Write the estimated probability as a decimal.

The planner calls a trip smooth when neither vehicle is late. Use the ten trials to estimate the probability of a smooth trip.

Estimated probability: %

The student runs 50 additional trials. Both vehicles are late in 9 of those additional trials. Combine these results with the original ten trials.

What is the combined estimated probability that both vehicles are late? Write a decimal.

Another student runs ten trials with the same correct model but gets no trips with both vehicles late. What is the best conclusion?

A. Both vehicles can never be late on the same trip.B. The subway must be on time whenever the bus is late.C. A small simulation can miss an event that is possible.D. The random-digit model must use different delay rates.

Use and Improve the Simulation

Use the stated simulation results for each question. Estimates describe patterns, not guaranteed outcomes.

10.

Nia uses ten cards labeled 0–9 instead of a random-digit generator. She draws the bus card and keeps it out while drawing the subway card. What change makes her procedure match the independent model?

A. She should return the bus card and shuffle before drawing again.B. She should remove every card labeled with an on-time digit.C. She should keep drawing until the subway card shows on time.D. She should use the bus card again as the subway card.

11.

The planner adds a time buffer whenever at least one vehicle is late. Use this larger simulation to estimate how often a buffer is needed.

Results of 200 Simulated Trips
Outcome	Number of trips
Neither late	108
Only bus late	42
Only subway late	34
Both late	16

Estimated percentage needing a buffer: %

12.

Use the 200-trip simulation above. Eli plans to make 50 bus-and-subway trips. About how many of those trips would you predict will have both vehicles late?

Predicted number: trips

13.

The planner wants a more stable estimate of the probability that both vehicles are late. Which approach is best?

A. Stop running trials as soon as both vehicles are late once.B. Repeat ten-trial batches and keep only the batch with the most delays.C. Change the late digits whenever a batch has no delays.D. Run many more independent trials and include all their results.

14.

A different itinerary uses three subway rides, each with a 20% chance of being late. Assume the rides are independent. Design a random-digit simulation to estimate the probability that at least two rides are late. Explain what the digits represent, how many digits make one trial, what counts as a success, and how you would calculate the estimate after 100 trials.