The following bus schedule is seen at a bus stop located somewhere in between town A and town B.
Town A-00:10, then every 20 mins
Town B-00:15, then every 20 mins
If a person arrives at this bus stop at some random time, the probability that the next bus is for town B is
Bus Schedule Probability: Town B Next Bus
This problem requires calculating the probability that the next bus encountered at a random arrival time is heading towards Town B, given specific schedules for Town A and Town B.
Bus Schedules and Departure Intervals
The bus schedules provided are:
- Town A: Departs at 00:10, followed by departures every 20 minutes (e.g., 00:10, 00:30, 00:50, ...).
- Town B: Departs at 00:15, followed by departures every 20 minutes (e.g., 00:15, 00:35, 00:55, ...).
To find the probability, we first determine the time intervals between consecutive bus departures from the stop, regardless of their destination.
Listing departures chronologically within a cycle:
- 00:10 (Town A)
- 00:15 (Town B)
- 00:30 (Town A)
- 00:35 (Town B)
- 00:50 (Town A)
- 00:55 (Town B)
Calculating the time gaps between these consecutive departures reveals a pattern:
- 00:10 to 00:15 = 5 minutes (Next bus: Town B)
- 00:15 to 00:30 = 15 minutes (Next bus: Town A)
- 00:30 to 00:35 = 5 minutes (Next bus: Town B)
- 00:35 to 00:50 = 15 minutes (Next bus: Town A)
- 00:50 to 00:55 = 5 minutes (Next bus: Town B)
- 00:55 to 01:10 = 15 minutes (Next bus: Town A)
This pattern of 5-minute and 15-minute intervals repeats every 20 minutes. The total duration of these intervals within one cycle is 5 + 15 = 20 minutes.
Probability Calculation for Town B
Since the person arrives at a random time, the probability of arriving during a specific interval is proportional to the length of that interval.
The total time within the 20-minute cycle where the *next* bus is for Town B corresponds to the sum of the 5-minute intervals:
Total time for next bus to be Town B = 5 minutes.
The probability is calculated as the ratio of the favorable interval lengths (next bus is B) to the total interval length within the cycle:
$ P(\text{Next bus is for Town B}) = \frac{\text{Total interval duration for next bus to B}}{\text{Total cycle interval duration}} $
$ P(\text{Next bus is for Town B}) = \frac{5 \text{ minutes}}{20 \text{ minutes}} $
$ P(\text{Next bus is for Town B}) = \frac{1}{4} $
Conclusion on Bus Probability
The probability that the next bus the person catches is for Town B is 1/4.
Important Questions from Probability (Notes)
- A box contains 20 black, 22 white, and 24 red socks. If a person draws socks at random one by one without looking, what is the minimum number of socks she must pick to be certain of having at least one pair of black socks?
- Some, but not all, faces of a six-faced cubical fair die are painted red (R) and the remaining green (G); and the die is thrown until red faces come up on top 4 times.
Consider the following sequences of colours listed left to right as they appear on the top.
A: GRRRR
B: GRGRRR
Which one of the following is true? - In a class, 40% and 20% students passed in Mathematics and Physics, respectively, and 10% students passed in both subjects. What is the probability of a randomly selected student to have passed in Physics if the student already passed in Mathematics?
- A stick of length L is broken into two pieces at random. What is the average length of the smaller piece?
- Two students are solving the same problem independently. If the probability that the first one solves the problem is $\frac{3}{5}$ and the probability that the second solves the problem is $\frac{4}{5}$, what is the probability that at least one of them solves the problem?
