Math nerds

[yh125d]

My brother approached me with an interesting problem, which I've been trying to solve. I've never been any good with probabilities so it's giving me a bit of trouble. I thought some of you might be interested

---------------
Santa Claus leaves his girlfriend's house drunk on Christmas Eve, and
walks outside toward his sleigh.

Santa leaves at 1:00 a.m., and he is drunk so he is only able to make 1
step forward OR 2 steps backward every 10 minutes. Santa goes hard, I'm
tellin you.

There is a 1/T chance that he will stumble 2 steps backwards, otherwise
he will move forward 1 step.

T = the current hour, rounded down. i.e. if it is 4:12 am, then T = 4.
If it is 5:56 am, T = 5. Assume Military time past noon, so 1:38 p.m.
would be T = 13.

Santa's sleigh is 30 steps away. What time is statistically most likely
for Santa to reach his sleigh?

---------------


Things I've clarified with him is that the first step is attempted at 1:10, and he can never be at a negative value, meaning he can't step back past the door



I was well on my way to solving it til he clarified that he cannot step farther back than the door, which complicated it a lot for me. It's been a couple years since my last calc class so I'm rusty. To be honest I don't even know that he has the correct answer himself, I doubt he can solve this problem

 

[xSauronx]

meh

im gonna go play nhl 10

 

[destrekor]

meh

im gonna go play nhl 10

And I'm going to attempt to work on my Russian presentation.

I'll be here again in 5 minutes. Possibly less.

Possibly now.

 

[Mike Gayner]

Ugh statistics ><

 

[Syringer]

n/m

 

[Mike Gayner]

Um is this a trick question? If T starts out at 1, there's a 1/1 chance that he'll walk backwards. There's no scenario that he'll walk forward if he starts at 1am.

...until the time is 2am.

 

[Syringer]

@Mike yup I'm dumb

I'm going to create a spreadsheet and calculate his expected values here.

 

[Mike Gayner]

@Mike yup I'm dumb

I'm going to create a spreadsheet and calculate his expected values here.

I almost came to the same conclusion...

PS: I failed stats so don't ask me to actually solve this.

 

[Gibson486]

Answer:

He won't. As stated, "Santa goes hard, I'm tellin you."

Therefore, he went back to his gf's house and did one more round. He then slept over only to wake up to Mrs. Clause with a knife in her hand.

 

[destrekor]

10:30
?

I went a very non-mathematical route, as statistically/probability rules that every new moment to take a step is a new moment, the previous and next having no weight.
And to calculate that accurately, I don't possibly care to do. Nor do I likely have the true math skills.
I took a statistics course on Probability. I could possibly do this with the tools they taught me, if I remembered them. And digging that notebook up does not sound like a good idea.

Mainly because I'm supposed to be working on a Russian presentation. I need to do it.
Otherwise I probably would.

edit:
all I can remember, is it sounds like ANDing? Compound-anding? If I suggest anything further, it'll have a 75&#37; chance of resembling BS.

 

[yankeesfan]

edit: let me think about this again.

 

[Syringer]

No clue if I did it right, but here is my effort.

steps in 10 minutes= 2(1/T)+1(1-(1/T))
steps every hour = 6* 2(1/T)+1(1-(1/T)) = 6+6/T

overall steps function = (6+6/T)*T

60=(6+6/T)*T
T= 9

9 hr +1:00am = 10:00am

Why is it 60? It's 30 steps yeah?

And I don't see how this accounts if he stumbles backwards when he starts out.

 

[yankeesfan]

Why is it 60? It's 30 steps yeah?

And I don't see how this accounts if he stumbles backwards when he starts out.

You are right. It is 30 steps.

This problem has more to do with probability than I thought. It can't be solved like I tried.

 

[DrPizza]

1 step every 10 minutes? So, should we assume steps are 10 minutes apart? Or that the mean difference in time between steps is 10 minutes. And, if the mean difference in time between steps is 10 minutes, what's the standard deviation? Since we're lacking that information, I'm going to assume 1 step every 10 minutes. And, since we're starting at 1:00, then the times of the next step forward (or two steps back) will be at 1:10. Then 1:20.

So, wtf does 5:56, 4:12, or 1:38 have to do with anything. And, which way should be round 1:30?

 

[halik]

Define "most likely"?

The description above is a Markov Chain stochastic process based on a changing binomial distribution, so if you're looking for the average outcome you probably have to run a montecarlo on it.

Edit:
After thinking about it for a bit, I don't think there's any direct solution for what he's asking. You can come up with the expected value of the step, but not to say anything about 30 steps fwd.

He could get to the sled in 300 + 50 mins (1/T = 1/1am = p(2 steps back| 1am) = 1) and then one of the paths of the process would be fwd-fwd-fwd-fwd ... etc.

So you simulate the process 10K times or something and then compute the mean number of steps taken to go 30 fwd.

 

[Born2bwire]

Off-hand, I'd say something like 11:49 AM. Don't know why you bother to start at 1 AM since the probability is P = 1/T where T = floor(time in hours). So he isn't going to move forward at all until 2 AM.

Define "most likely"?

The description above is a Markov Chain stochastic process based on a binomial distribution, so if you're looking for the average outcome you probably have to run a montecarlo on it.

We could do a random walk. :awe:

Serously though this is probably the proper way of treating it. This is sort of like what I have done but I am not doing the smart thing of actually doing a monte carlo sampling.

 

[yankeesfan]

Define "most likely"?

The description above is a Markov Chain stochastic process based on a binomial distribution, so if you're looking for the average outcome you probably have to run a montecarlo on it.

Awesome. So it is unsolvable by hand?

 

[yh125d]

I worked out a very rough estimate of 12:50. Bro says it's 11:40, and he solved with really simple excel formulas which smells like bullshit. I may not know how to properly and accurately solve this but I know for damn sure it's not a simple probability question he could do in excel in 5 minutes. When I questioned him on this he said he got the same answer as thousands of other nerds


Halik, that is along the lines of what I was thinking, just haven't been able to put it into words

 

[halik]

I worked out a very rough estimate of 12:50. Bro says it's 11:40, and he solved with really simple excel formulas which smells like bullshit. I may not know how to properly and accurately solve this but I know for damn sure it's not a simple probability question he could do in excel in 5 minutes. When I questioned him on this he said he got the same answer as thousands of other nerds


Halik, that is along the lines of what I was thinking, just haven't been able to put it into words

Actually I suppose you can count up the expected steps:

at 2am the expected step is -2/2 + 1/2 = -1.5, so he won't be going anywhere
at 3am the expected step is 0, so again no movement
at 4am the expected step is -2/4 + 1*3/4 = 1/4 and there's 5 of em = 1/4 *5 = 1.25
at 5am the expected step is -2/5 + 1*4/5 = 2/5 -//- = 0.4*5 + 1.25 = 3.25
and iterate till your total hits 30

 

[yh125d]

Actually I suppose you can count up the expected steps:

at 2am the expected step is -2/2 + 1/2 = -1.5, so he won't be going anywhere
at 3am the expected step is 0, so again no movement
at 4am the expected step is -2/4 + 1*3/4 = 1/2 and there's 5 of em = 1/2 *5
and interate till your total hits 30

Thats pretty much how I went about it.

At 5AM, he's +1.5
At 6AM, +3.9
7 - 6.9
8 - 10.329
9 - 14.079
10 - 18.079
11 - 22.279
12 - 26.629

And should hit 30 at my guess of 12:50


Very rough, but I don't think it'd be 1:10 longer than the "right" answer (pretty sure there isn't an answer as you said).

 

[Born2bwire]

I worked out a very rough estimate of 12:50. Bro says it's 11:40, and he solved with really simple excel formulas which smells like bullshit. I may not know how to properly and accurately solve this but I know for damn sure it's not a simple probability question he could do in excel in 5 minutes. When I questioned him on this he said he got the same answer as thousands of other nerds


Halik, that is along the lines of what I was thinking, just haven't been able to put it into words

I did it in Matlab and it doesn't take more than a few minutes. Let's see if code works...

time = 1;
step = 1/6;
startdist = 0;
enddist = 30;
dist = startdist;
sumtime = 0;
iter = 500000;

for n=1:iter
    while dist < enddist
        time = time+step;
        prob = 1.0/floor(time);
        if (rand() <= prob)
            dist = dist-2;
        else
            dist = dist+1;
        end
        if (dist < startdist)
            dist = startdist;
        end
    end
    sumtime = sumtime+time/iter;
    time = 1;
    dist = startdist;
end
averagetime = sumtime;

hour = floor(averagetime);
minute = floor((averagetime-hour)*60);
second = floor((averagetime-hour-minute/60)*60^2);

display(hour); display(minute); display(second);

This isn't too different from the Markov chain that was previously suggested since we are restricting movement along one dimension. However, I do a brute force method as opposed to using an appropriate sampling, like Monte Carlo, to estimate the answer. Because this is so numerically simple I do not see any reason wrong with doing brute force though the convergence is poor such that I can only state up to the minute of arrival and not to the second.

I do not think that the expectation value of each hourly interval is going to work correctly. It completely negates any forward movement for the first two hours that it is possible. I think a Bayesian analysis would be more appropriate as that should still yield a net forward movement for the hours of 2 and 3 AM.

What I mean is, let's say that there is a 1&#37; chance of moving forward during 3-3:50 AM. Your expectation value gives a zero expected movement but if we allow the 1% chance, then we start out at 4 AM with an additional 0.01 distance. The error in your analysis is that you still allowed for negative movement during the one hour interval. Instead, you should do a Bayesian analysis for each time step. For example:

2 AM: 50% chance of moving forward

This branches out to:

2:10 AM: 1) Dist = 0 (50%) or 2) Dist = 1 (50%)

And then you branch out here which is essentially what the Markov chain is supposed to estimate numerically.

 

[chuckywang]

This one requires MATLAB, especially with that starting condition.

 

[iCyborg]

I wrote a C++ program to simulate 1,000,000 iterations and I get around 11:44-11.45, since that's not a valid step, 11:40 could be the most likely time.
No guarantees that I didn't f*ck up something in the code though...

 

[Cogman]

You have to define most likely. Are we talking 90&#37; chance, 80% chance, 51% chance. It makes a huge difference in what time you say he would most likely arrive. I could say 1am next Tuesday he will most likely arrive, however, if likely is 99.999999999999999999999% then he won't 'most likely' arrive by next Tuesday. If you say 100%, then he will never most likely arrive.

I should note that any calculation or method of solving this is moot without a firm definition of most likely.

 

[iCyborg]

The probability that he will reach the sleigh at 1:00 is 0, 1:10 0, etc., for each time you can compute the probability that he will reach the sleigh at that time (and not have been there earlier). The most likely time is the time that has the biggest probability. I don't think there's anything moot here.