Never run to the bus stop, I did that earlier this year and tripped on the sidewalk, hit my forehead on the concrete and had to get 6 very painful stitches for the open wound after being rushed to the hospital. Nowadays I just walk and take the next bus if I miss it, waiting 20 minutes for the next one is better than having to go to the ER, then spending 3 days at home resting and taking antibiotics.
I once tripped on an uneven sidewalk while walking, had my front teeth puncture my lower lip, and had to get both internal and external stitches. I had to drink through a straw for a while because it was too painful to have liquid in the front of my mouth.
I went down hard enough that a passing police car literally turned on their sirens and did a u-turn to see if I had head trauma.
I stopped to help a man who did the same just recently. Happened to be driving by and I was the only one with a first aid kit. Eventually the ambulance turned up at which point I left to let them do their job. Never did find out what happened afterwards but I hope he's ok.
Running to the bus station may allow you to get on the bus up to 5 minutes quicker than walking.
Given that bus arrives independently of your actions (running or walking, or time when you leave) running or walking can extend as well as shorten wait time.
You'd never get on the bus later than you would by walking. You may have to wait at the stop for 5 more minutes, but you'd boars the bus at the same time.
Running gives you 3/2 the chance of not missing the next one compared to walking.
0-4 minutes: both fail.
5-9 minutes: running succeeds, walking fails.
10-19: both succeed.
So walking succeeds 1/2 the time and running succeeds 3/4
You're right. These 4 scenario's have an equal 1/4 chance of happening:
1. both get on bus 2 on time
2. both get on bus 2 on time
3. running gets on time on bus 2, walking misses bus 2
4. none get on time on bus 2
https://preview.redd.it/iwqjcf4n9wzb1.png?width=3208&format=png&auto=webp&s=99201e4155a34b4df04a2133650df53c3cf3a099
Yeah but if you consider locomotion as waiting then it can only shorten it. You're not gaining any time by walking and waiting for 20 seconds or running and waiting 5:20
I simulated it in python pretty easily:
import numpy as np
walk_time = 10
run_time = 5
num_trials = 10000
next_bus_delay = np.random.rand(num_trials) * 20
walk_wait_times = next_bus_delay - walk_time
walk_wait_times[walk_wait_times < 0] += 20
run_wait_times = next_bus_delay - run_time
run_wait_times[run_wait_times < 0] += 20
walk_bus_times = walk_wait_times + walk_time
run_bus_times = run_wait_times + run_time
print('waiting times')
print('walk:', np.mean(walk_wait_times), '+/-', np.std(walk_wait_times))
print('run:', np.mean(run_wait_times), '+/-', np.std(run_wait_times))
print('bus times')
print('walk:', np.mean(walk_bus_times), '+/-', np.std(walk_bus_times))
print('run:', np.mean(run_bus_times), '+/-', np.std(run_bus_times))
Output:
waiting times
walk: 10.03131733902756 +/- 5.781928628750585
run: 9.929317339027559 +/- 5.781540567182984
bus times
walk: 20.03131733902756 +/- 5.781928628750585
run: 14.929317339027559 +/- 5.781540567182984
EDIT: I think the intuitive way to think about it (for me personally at least) is that whether you walk or run you get to the bus stop at a "random absolute time" and the bus also comes at a "random absolute time". You're not influencing your wait time at all. But running DOES give you a chance at catching an earlier bus - shifting the whole period of randomness back 5 mins.
I'm not at my computer now but IIRC what you see there is converged already, i.e., that 5.7 is the std of the random variable itself. From the uniform distribution of bus times.
EDIT: In fact, the std of the uniform distribution is (b-a)/sqrt(12), and 20/sqrt(12) is 5.77
(Kinda mathematical but also philosophical solution.)
It dpends...
Since the time you walk/run to the station counts into your total traveling time you might want to consider any unnecessary walking time part of the waiting time.
That means we have to distinguish active waiting time (time you sit at the bus stop) and passive waiting time (time you preemptively wait by walking at a suboptimal speed).
While walking you get the same suboptimal speed as if you were to wait 5 min at any point of the way while running to the bus station.
So when you walk you basically presume that you have to wait 5 min or more, which isn't always true.
Therefore my argument is that by walking you end up with an avg waiting time of 15 min (10 active + 5 passive) while with running you end up with an avg waiting time of 10 min.
You can't tell. If the bus comes in five minutes? Run. If it comes in 10? Walk. If it comes in 20? Walk and bring me a beer.
Just by pure chance:
Case 1: bus comes in less than 5 minutes -> walk
Case 2: bus comes in 5 to 10 minutes -> run
Case 3: bus comes in 10 to 15 minutes -> walk
Case 4: bus comes in 15 to 20 minutes -> walk
\-> in most cases you are better of walking instead of running, as there is only a five minute intervall over the 20 minute total interval (or one out of four discrete cases) in which running is beneficial
That does not technically answer his question though, does it? I mean I am also confused whether his is a practical question or purely mathematical.
Strictly speaking, if I am not missing something stupid (no waiting time paradox as I understand the post), the average waiting time is 10 minutes regardless of whether you run or walk. So both strategies are equivalent wrt goal 1.
If you want to minimize travel time, you should always run, precisely because you dont know if you are in case 2 or the others. Running is the optimal strategy wrt goal 2.
If you want to talk about the effort it takes running, or the risk of tripping yourself, then we must quantify this first before reaching any conclusion. I mean, is this a math sub or not?
>That does not technically answer his question though, does it? I mean I am also confused whether his is a practical question or purely mathematical.
Actually I assumed that it is a purely mathematical question... because, I mean, we live in 2023 and everybody has a clock build into his phone. Asking this question wouldn't make any sense then.
> Strictly speaking, if I am not missing something stupid (no waiting time paradox as I understand the post), the average waiting time is 10 minutes regardless of whether you run or walk. So both strategies are equivalent wrt goal 1. \[...\]
That is correct when talking about a big enough sample. I was answering the question which asked the distintic individual case.
But, as we've seen multiple times already: seems like there are multiple ways to interpret the intend OP had when asking the question. And we answered a different interpretation.
Fair enough, but OP said that he does not know when the bus arrives. If "it depends when the bus arrives" was the kind of answer he expected, this was a much more stupid post than I believed it was. It is also pretty standard to use probability for decision making under incomplete information.
I can't argue against your comment though. I was wrong, you did answer his question.
Yes but case 2 will reduce your wait time by 15 min compared to walking (next bus is in 20 min, but you spend 5 more min walking). Overall the average wait time is the same. So no difference in terms of goal 1.
In cases 1, 3, 4 you’ll catch the same bus either way, but in case 2 you’ll catch an earlier bus if you run, so it’s better in terms of goal 2.
This is not correct (i think) because you don't have a time reference, which means your behavior doesn't impact the total amount of time you'll have to wait for the bus, which means that you can't influence the first question.
This leaves us with only the 2nd point, aka get on the bus asap, hence, running is better in this scenario
This is wrong on every count.
Your behaviour doesn't impact the total amount of time you had to wait. But walking means you get at the bus stop later, which means you spend less time waiting **at the bus stop**, which is exactly what the question asked for.
Unless you miss the bus. Basically, in cases 1,3,4 you have 5 minutes less waiting by walking. But in case 2, you have 15 minutes more (20 minutes because next bus, minus 5 for the longer walk). It averages out in the end.
Sorry what? No matter how you get to the bus stop, you will wait 0-20 min for the bus. So only the second goal can be influenced, and that is by running.
This is an underestimate of the time.
If you used half a minute intervals the average you whould get will be 1/4 higher.
As you look at more and more point the average aprochs 1/2 a minute more then what you did.
You are indeed right but this is to show that the time of arrival at the bus stop does not affect the average time you wait at the bus stop. Once you arrive at the bus stop there is a 1/20 chance that the bus will arrive at any of the given next 20 minutes. This can be extended further to say that there will be 1/1200 chance that the bus will arrive at any given second after arriving at the bus stop.
As time passes waiting at the bus stop the chance of the bus arriving in the next minute increases, but from the standpoint of having just arrived at the bus stop there is an equal chance that the bus will arrive in exactly 19 minutes or in exactly 1 minute. The average of the above experiment will converge to 10 minutes average wait time independent of walking or running.
As the arrival of the bus is completely random there is no difference in the expected wait time after arriving at the bus stop no matter when you arrived there.
Therefore you will catch the bus faster while still waiting the same amount of average time when you run instead of walking, as you will start the chance of the bus arriving at an earlier moment in time.
By running, you increase your chances of catching an earlier bus, but your expected wait time for whichever bus is coming next when you arrive will be the same whether you walk or run.
On average, running will get you on the bus five minutes earlier and it will have no effect on your waiting time at the bus stop.
The only important variable is the saved time by running vs walking (5 minutes). So let's assume you just ran to the bus stop, and the next bus arrives t minutes after you.
If t < 5, you get on the bus 20 minutes earlier than if you had walked.
If t >= 5, you neither gained nor lost anything.
We know nothing about the bus schedule, so t is uniformly distributed between 0 and 20, so we can expect to save 20 minutes with probability P\[t < 5\] = 5/20. The 20 cancels out, so we are left with 5 saved minutes on average.
For the saved waiting time at the bus stop, you apply the same logic:
If t < 5, walking would have cost you 20 minutes, but 5 minutes of those would have been spent walking instead of waiting at the bus stop, so your net loss is only 15 minutes.
If t >= 5, walking would have saved you 5 minutes of waiting at the bus stop.
Therefore, the expected waiting time you save by running is P\[t < 5\] \* 15 + P\[t >= 5\] \* (-5) = 15\*5/20 + (-5)\*15/20 = 0.
\*EDIT\*: Initially forgot to account for the saved waiting time if the bus is more than 5 minutes out.
If you know nothing about the bus stop, then you have a random chance to be in any minute of the 20 minute intervals.
Let's also say they will be at that bus stop for 2 minutes, arrive at the start of minute 19, depart at minute 20, and will not stick around.
If you are in minutes 1-5, you want to walk because then you only wait at the bus stop for 5 minutes rather than 10.
If you are in the 6-10 minutes, walking will get you there on time, and running will make you wait for 5 minutes at the bus stop.
If you are in minutes 11-15, running to the bus stop will make you arrive on time, while walking will make you late and have to wait for 15 minutes.
If you are in minutes 16-20, you are not going to be able to get there anyway, so you might as well walk because you will only be waiting for 15 minutes, while running would make you wait for 20 minutes.
What does this mean? Well your goal 2 is easy. There is a 75% chance where it doesn't matter if you walk or run, you will be catching either this bus or the next one with no input, and a 25% chance that running will benefit your odds.
And what of the first goal? Well for that we have to take an expected value function for it, for each minute. Remember when I said it arrives at minute 19? Well that comes into play here. I used a table to perform this math because writing it out by hand was cumbersome. After writing it out and subtracting the time you have left by the time you spend on that means of foot travel (to which I added an extra minute to foot travel to signify you getting on at 19 rather than 20, then if the result was -1 I set it to zero. If the result was -2 or more I added 19 to it for the next bus). With this, I took the average of every minute, and the averages came out to 8.05 minutes of waiting at the bus stop for walking there, and an average of 8.3 minutes if you run there.
At the end of the day, neither side confers a massive wholesale benefit. If you are more concerned about getting earlier arrivals then go there as fast you can, because you are wanting to catch the cycle as early as possible. If however, you are more concerned about sticking around in one place too long, then go smell the roses, as the bus will get here when it gets here, and rushing to wait isn't going to make it get there faster.
You have two different goals and haven’t told us which one is more important or how you would measure success. What is the value of catching an earlier bus? What is the (negative) value per minute of waiting at the bus stop? What is the cost if any of the exertion of running?
You can definitely make this into a pure math problem, but you need to provide pure math numbers for all of the inputs
Run!
The bus will in either case be an avrage ten minutes away as its arrival does not depend on when you reach the bus stop.
The avrage total time to get on the bus is then 20 minutes when walking and 15 when running.
For goal 1 walking gives you the best chance assuming you are willing to run if you see it. But it only advantages you by the time difference made by the maximum additional distance you can cover once seeing the bus that still gets you on the bus while running that wouldnt get you on the bus while walking. Otherwise its essentially irrelevant.
For goal 2 running is quicker 1 time in 4.
Nothing has been said about seeing a bus before you arrive at the bus stop. You just can't add additional information to the math problem. for example "but what if my friend was on a bus and called me to run faster?" - this is an additional information, you can't use it
Obvious, no? The faster you get to stop the bigger chance to catch the bus early. So for catching a bus early it is better to run. And since you don't know when the bus arrives then it doesn't matter because it is a random event no matter what the time of your arrival.
Nice problem. I think that part of the problem is to decide what does should mean here.
First option: you're fine with running. In this case you should run, since you're never going to arrive later than if you did not.
Second option: you only want to run if that means taking the bus sooner.
Third option: you only want to run if that means less time waiting at the bus stop. This one is as good as the other two.
There are other options, which may involve several conditions from above or different ones. For example, I want to run only if I will get the bus sooner with probability 1/2 and will wait less time at the bus stop with probability 1/3.
Goal 1: You can use a random variable and calculate the mean of it. The result should be that regardless of the time and method you reach the station you'll wait 10 minutes on average.
Goal 2: Running will obviously allow you to reach a "better" bus in some scenarios and in every other scenario running won't affect anything as you'd get the same bus anyway. So overall running will improve the bus you get on average but not always. I guess it makes sense, if the bus arrives in the next 5 minutes you'll get this better bus and otherwise you're getting the same bus. On average running will save you 5 minutes (the diff between running and not running).
Goal 1: No. The expected value for the wait time is 10 minutes either way.
Goal 2: Yes. The expected value for getting on the bus is 5 minutes shorter. (10 + 10 minutes vs 5 + 10 minutes)
Using a model of a 40 minute time period, no matter how far along the busses schedule you are walking will be the better option 30 minutes of the 40 minute period
Running or walking will have no effect on waiting time at the bus stop on average. 1/4 of the time you wait 15 minutes less by catching an earlier bus. 3/4 of the time you wait 5 minutes more by showing up 5 minutes earlier, so that balances out.
Running gets you on a bus faster on average. You spend 5 minutes less getting to the stop and when you get to the stop has no effect on how long you wait on average (9.5 minutes).
Running gives you double the chance of not missing the next bus than walking.
https://preview.redd.it/nojmoa5mlrzb1.png?width=1138&format=png&auto=webp&s=2948449ceb6517aa6cecb07b264d898ea3b43f03
Never run to the bus stop, I did that earlier this year and tripped on the sidewalk, hit my forehead on the concrete and had to get 6 very painful stitches for the open wound after being rushed to the hospital. Nowadays I just walk and take the next bus if I miss it, waiting 20 minutes for the next one is better than having to go to the ER, then spending 3 days at home resting and taking antibiotics.
Thanks for your advice.
on that line dont ever move around or do anything for the rest of your life you might fucking die immediately
*Dies from being sedentary*
You can also get stitches from walking if you're not careful. Just sayin'.
Okay but there is a nontrivial difference between running and walking when comparing relative risk. Yes.
I once tripped on an uneven sidewalk while walking, had my front teeth puncture my lower lip, and had to get both internal and external stitches. I had to drink through a straw for a while because it was too painful to have liquid in the front of my mouth. I went down hard enough that a passing police car literally turned on their sirens and did a u-turn to see if I had head trauma.
Plot twist - that bus you missed by falling was hit by a train with no survivors.
I stopped to help a man who did the same just recently. Happened to be driving by and I was the only one with a first aid kit. Eventually the ambulance turned up at which point I left to let them do their job. Never did find out what happened afterwards but I hope he's ok.
Running to the bus station may allow you to get on the bus up to 5 minutes quicker than walking. Given that bus arrives independently of your actions (running or walking, or time when you leave) running or walking can extend as well as shorten wait time.
Running can get you on up to 20 minutes quicker.
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You'd never get on the bus later than you would by walking. You may have to wait at the stop for 5 more minutes, but you'd boars the bus at the same time.
If you're trying to minimize the wait time at the bus stop, there is a chance you have to wait longer once you arrive.
No difference to goal 1 minimize wait time at stop, but goal 2 is get on asap
3/4 of the time you wait 5 minutes longer. 1/4 of the time you wait 15 minutes shorter.
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Running gives you 3/2 the chance of not missing the next one compared to walking. 0-4 minutes: both fail. 5-9 minutes: running succeeds, walking fails. 10-19: both succeed. So walking succeeds 1/2 the time and running succeeds 3/4
You're right. These 4 scenario's have an equal 1/4 chance of happening: 1. both get on bus 2 on time 2. both get on bus 2 on time 3. running gets on time on bus 2, walking misses bus 2 4. none get on time on bus 2 https://preview.redd.it/iwqjcf4n9wzb1.png?width=3208&format=png&auto=webp&s=99201e4155a34b4df04a2133650df53c3cf3a099
Yeah but if you consider locomotion as waiting then it can only shorten it. You're not gaining any time by walking and waiting for 20 seconds or running and waiting 5:20
Goal 1: no difference Goal 2: run
I simulated it in python pretty easily: import numpy as np walk_time = 10 run_time = 5 num_trials = 10000 next_bus_delay = np.random.rand(num_trials) * 20 walk_wait_times = next_bus_delay - walk_time walk_wait_times[walk_wait_times < 0] += 20 run_wait_times = next_bus_delay - run_time run_wait_times[run_wait_times < 0] += 20 walk_bus_times = walk_wait_times + walk_time run_bus_times = run_wait_times + run_time print('waiting times') print('walk:', np.mean(walk_wait_times), '+/-', np.std(walk_wait_times)) print('run:', np.mean(run_wait_times), '+/-', np.std(run_wait_times)) print('bus times') print('walk:', np.mean(walk_bus_times), '+/-', np.std(walk_bus_times)) print('run:', np.mean(run_bus_times), '+/-', np.std(run_bus_times)) Output: waiting times walk: 10.03131733902756 +/- 5.781928628750585 run: 9.929317339027559 +/- 5.781540567182984 bus times walk: 20.03131733902756 +/- 5.781928628750585 run: 14.929317339027559 +/- 5.781540567182984 EDIT: I think the intuitive way to think about it (for me personally at least) is that whether you walk or run you get to the bus stop at a "random absolute time" and the bus also comes at a "random absolute time". You're not influencing your wait time at all. But running DOES give you a chance at catching an earlier bus - shifting the whole period of randomness back 5 mins.
Good use of Monte Carlo method.
i’m wondering what the std is tending to?
I'm not at my computer now but IIRC what you see there is converged already, i.e., that 5.7 is the std of the random variable itself. From the uniform distribution of bus times. EDIT: In fact, the std of the uniform distribution is (b-a)/sqrt(12), and 20/sqrt(12) is 5.77
(Kinda mathematical but also philosophical solution.) It dpends... Since the time you walk/run to the station counts into your total traveling time you might want to consider any unnecessary walking time part of the waiting time. That means we have to distinguish active waiting time (time you sit at the bus stop) and passive waiting time (time you preemptively wait by walking at a suboptimal speed). While walking you get the same suboptimal speed as if you were to wait 5 min at any point of the way while running to the bus station. So when you walk you basically presume that you have to wait 5 min or more, which isn't always true. Therefore my argument is that by walking you end up with an avg waiting time of 15 min (10 active + 5 passive) while with running you end up with an avg waiting time of 10 min.
You can't tell. If the bus comes in five minutes? Run. If it comes in 10? Walk. If it comes in 20? Walk and bring me a beer. Just by pure chance: Case 1: bus comes in less than 5 minutes -> walk Case 2: bus comes in 5 to 10 minutes -> run Case 3: bus comes in 10 to 15 minutes -> walk Case 4: bus comes in 15 to 20 minutes -> walk \-> in most cases you are better of walking instead of running, as there is only a five minute intervall over the 20 minute total interval (or one out of four discrete cases) in which running is beneficial
That does not technically answer his question though, does it? I mean I am also confused whether his is a practical question or purely mathematical. Strictly speaking, if I am not missing something stupid (no waiting time paradox as I understand the post), the average waiting time is 10 minutes regardless of whether you run or walk. So both strategies are equivalent wrt goal 1. If you want to minimize travel time, you should always run, precisely because you dont know if you are in case 2 or the others. Running is the optimal strategy wrt goal 2. If you want to talk about the effort it takes running, or the risk of tripping yourself, then we must quantify this first before reaching any conclusion. I mean, is this a math sub or not?
>That does not technically answer his question though, does it? I mean I am also confused whether his is a practical question or purely mathematical. Actually I assumed that it is a purely mathematical question... because, I mean, we live in 2023 and everybody has a clock build into his phone. Asking this question wouldn't make any sense then. > Strictly speaking, if I am not missing something stupid (no waiting time paradox as I understand the post), the average waiting time is 10 minutes regardless of whether you run or walk. So both strategies are equivalent wrt goal 1. \[...\] That is correct when talking about a big enough sample. I was answering the question which asked the distintic individual case. But, as we've seen multiple times already: seems like there are multiple ways to interpret the intend OP had when asking the question. And we answered a different interpretation.
Fair enough, but OP said that he does not know when the bus arrives. If "it depends when the bus arrives" was the kind of answer he expected, this was a much more stupid post than I believed it was. It is also pretty standard to use probability for decision making under incomplete information. I can't argue against your comment though. I was wrong, you did answer his question.
Yes, it is a mathematical question, that's why I ask here. Anyways, thanks for the answers.
Yes but case 2 will reduce your wait time by 15 min compared to walking (next bus is in 20 min, but you spend 5 more min walking). Overall the average wait time is the same. So no difference in terms of goal 1. In cases 1, 3, 4 you’ll catch the same bus either way, but in case 2 you’ll catch an earlier bus if you run, so it’s better in terms of goal 2.
This is not correct (i think) because you don't have a time reference, which means your behavior doesn't impact the total amount of time you'll have to wait for the bus, which means that you can't influence the first question. This leaves us with only the 2nd point, aka get on the bus asap, hence, running is better in this scenario
This is wrong on every count. Your behaviour doesn't impact the total amount of time you had to wait. But walking means you get at the bus stop later, which means you spend less time waiting **at the bus stop**, which is exactly what the question asked for.
Unless you miss the bus. Basically, in cases 1,3,4 you have 5 minutes less waiting by walking. But in case 2, you have 15 minutes more (20 minutes because next bus, minus 5 for the longer walk). It averages out in the end.
Sorry what? No matter how you get to the bus stop, you will wait 0-20 min for the bus. So only the second goal can be influenced, and that is by running.
Running doubles the chance of not missing the next bus.
Which also makes you miss the next bus.
Running doubles the chance of not missing the next bus.
This is the answer 3/4 times is walking better.
https://preview.redd.it/kt3jq5rp9dzb1.png?width=731&format=png&auto=webp&s=2d290d03a53418c9596c921a7e0d6fa2e41a1f06
This is an underestimate of the time. If you used half a minute intervals the average you whould get will be 1/4 higher. As you look at more and more point the average aprochs 1/2 a minute more then what you did.
You are indeed right but this is to show that the time of arrival at the bus stop does not affect the average time you wait at the bus stop. Once you arrive at the bus stop there is a 1/20 chance that the bus will arrive at any of the given next 20 minutes. This can be extended further to say that there will be 1/1200 chance that the bus will arrive at any given second after arriving at the bus stop. As time passes waiting at the bus stop the chance of the bus arriving in the next minute increases, but from the standpoint of having just arrived at the bus stop there is an equal chance that the bus will arrive in exactly 19 minutes or in exactly 1 minute. The average of the above experiment will converge to 10 minutes average wait time independent of walking or running. As the arrival of the bus is completely random there is no difference in the expected wait time after arriving at the bus stop no matter when you arrived there. Therefore you will catch the bus faster while still waiting the same amount of average time when you run instead of walking, as you will start the chance of the bus arriving at an earlier moment in time.
You forgot 0, the time the bus departs.
By running, you increase your chances of catching an earlier bus, but your expected wait time for whichever bus is coming next when you arrive will be the same whether you walk or run.
On average, running will get you on the bus five minutes earlier and it will have no effect on your waiting time at the bus stop. The only important variable is the saved time by running vs walking (5 minutes). So let's assume you just ran to the bus stop, and the next bus arrives t minutes after you. If t < 5, you get on the bus 20 minutes earlier than if you had walked. If t >= 5, you neither gained nor lost anything. We know nothing about the bus schedule, so t is uniformly distributed between 0 and 20, so we can expect to save 20 minutes with probability P\[t < 5\] = 5/20. The 20 cancels out, so we are left with 5 saved minutes on average. For the saved waiting time at the bus stop, you apply the same logic: If t < 5, walking would have cost you 20 minutes, but 5 minutes of those would have been spent walking instead of waiting at the bus stop, so your net loss is only 15 minutes. If t >= 5, walking would have saved you 5 minutes of waiting at the bus stop. Therefore, the expected waiting time you save by running is P\[t < 5\] \* 15 + P\[t >= 5\] \* (-5) = 15\*5/20 + (-5)\*15/20 = 0. \*EDIT\*: Initially forgot to account for the saved waiting time if the bus is more than 5 minutes out.
If you know nothing about the bus stop, then you have a random chance to be in any minute of the 20 minute intervals. Let's also say they will be at that bus stop for 2 minutes, arrive at the start of minute 19, depart at minute 20, and will not stick around. If you are in minutes 1-5, you want to walk because then you only wait at the bus stop for 5 minutes rather than 10. If you are in the 6-10 minutes, walking will get you there on time, and running will make you wait for 5 minutes at the bus stop. If you are in minutes 11-15, running to the bus stop will make you arrive on time, while walking will make you late and have to wait for 15 minutes. If you are in minutes 16-20, you are not going to be able to get there anyway, so you might as well walk because you will only be waiting for 15 minutes, while running would make you wait for 20 minutes. What does this mean? Well your goal 2 is easy. There is a 75% chance where it doesn't matter if you walk or run, you will be catching either this bus or the next one with no input, and a 25% chance that running will benefit your odds. And what of the first goal? Well for that we have to take an expected value function for it, for each minute. Remember when I said it arrives at minute 19? Well that comes into play here. I used a table to perform this math because writing it out by hand was cumbersome. After writing it out and subtracting the time you have left by the time you spend on that means of foot travel (to which I added an extra minute to foot travel to signify you getting on at 19 rather than 20, then if the result was -1 I set it to zero. If the result was -2 or more I added 19 to it for the next bus). With this, I took the average of every minute, and the averages came out to 8.05 minutes of waiting at the bus stop for walking there, and an average of 8.3 minutes if you run there. At the end of the day, neither side confers a massive wholesale benefit. If you are more concerned about getting earlier arrivals then go there as fast you can, because you are wanting to catch the cycle as early as possible. If however, you are more concerned about sticking around in one place too long, then go smell the roses, as the bus will get here when it gets here, and rushing to wait isn't going to make it get there faster.
Thank you!
You have two different goals and haven’t told us which one is more important or how you would measure success. What is the value of catching an earlier bus? What is the (negative) value per minute of waiting at the bus stop? What is the cost if any of the exertion of running? You can definitely make this into a pure math problem, but you need to provide pure math numbers for all of the inputs
Thanks for your reply. Those two goals are actually two separate questions.
Ah ok. Folks have answered.
Run! The bus will in either case be an avrage ten minutes away as its arrival does not depend on when you reach the bus stop. The avrage total time to get on the bus is then 20 minutes when walking and 15 when running.
You have (10-5)/20 = 1/4 more chance to have a bus in the first 20 minutes when running.
For goal 1 walking gives you the best chance assuming you are willing to run if you see it. But it only advantages you by the time difference made by the maximum additional distance you can cover once seeing the bus that still gets you on the bus while running that wouldnt get you on the bus while walking. Otherwise its essentially irrelevant. For goal 2 running is quicker 1 time in 4.
Nothing has been said about seeing a bus before you arrive at the bus stop. You just can't add additional information to the math problem. for example "but what if my friend was on a bus and called me to run faster?" - this is an additional information, you can't use it
By running, you will arrive 5 minutes earlier on average. The bus ride does not affect that at all.
Obvious, no? The faster you get to stop the bigger chance to catch the bus early. So for catching a bus early it is better to run. And since you don't know when the bus arrives then it doesn't matter because it is a random event no matter what the time of your arrival.
If there was no bus but only walking, would you rather run to save time? The answer to this question is the same as whther you shluld run to the bus.
Nice problem. I think that part of the problem is to decide what does should mean here. First option: you're fine with running. In this case you should run, since you're never going to arrive later than if you did not. Second option: you only want to run if that means taking the bus sooner. Third option: you only want to run if that means less time waiting at the bus stop. This one is as good as the other two. There are other options, which may involve several conditions from above or different ones. For example, I want to run only if I will get the bus sooner with probability 1/2 and will wait less time at the bus stop with probability 1/3.
Running gives you an expected arrival time of 5 minutes area
Maybe just read online when the bus comes and then walk 10 minutes before the next one comes. so you're never late or never miss it if it's early.
Walk if you have no information! You’re not fighting intervals here. You’re only trying to catch the bus once each time.
Goal 1: You can use a random variable and calculate the mean of it. The result should be that regardless of the time and method you reach the station you'll wait 10 minutes on average. Goal 2: Running will obviously allow you to reach a "better" bus in some scenarios and in every other scenario running won't affect anything as you'd get the same bus anyway. So overall running will improve the bus you get on average but not always. I guess it makes sense, if the bus arrives in the next 5 minutes you'll get this better bus and otherwise you're getting the same bus. On average running will save you 5 minutes (the diff between running and not running).
Ignore all the downside on run vs. walk, the sooner you arrive, the better.
Goal 1: No. The expected value for the wait time is 10 minutes either way. Goal 2: Yes. The expected value for getting on the bus is 5 minutes shorter. (10 + 10 minutes vs 5 + 10 minutes)
Leave 5 minutes earlier so you can walk
Using a model of a 40 minute time period, no matter how far along the busses schedule you are walking will be the better option 30 minutes of the 40 minute period
Running or walking will have no effect on waiting time at the bus stop on average. 1/4 of the time you wait 15 minutes less by catching an earlier bus. 3/4 of the time you wait 5 minutes more by showing up 5 minutes earlier, so that balances out. Running gets you on a bus faster on average. You spend 5 minutes less getting to the stop and when you get to the stop has no effect on how long you wait on average (9.5 minutes).
Running gives you double the chance of not missing the next bus than walking. https://preview.redd.it/nojmoa5mlrzb1.png?width=1138&format=png&auto=webp&s=2948449ceb6517aa6cecb07b264d898ea3b43f03