Math nerds

[Born2bwire]

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.

That would be different than say the expectation value which is what I and others have tried to evaluate. For example, I could have a distribution where the probability is: y(1) = 0.7, y(2) = 0.2, y(3) = 0.1. In this case, under your definition, the most likely value is 1 since the probability is the highest here. However, the expected value is 1.4. Finding the time with the highest probability is a much more difficult problem as you each hour presents 64 possible states that you have to map into.

 

[iCyborg]

Yes, we should be looking for the mode of the prob. distribution, and not the expectation (which is what I was doing too, though I used it only to estimate the correct value).
I should rewrite my program to actually look for the number of times we had arrived at a particular time, it should be a better estimate, though I feel it won't be far from the expectation.

 

[chuckywang]

You have to define most likely. Are we talking 90% 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.

I think the problem is very well defined. What is the most likely time he will arrive at the sleigh?

"Arrive" means the first time he reaches his sleigh. For any time that you choose, you can calculate the probability that he first reaches his sleigh at that time. What time has the maximum probability?

 

[Born2bwire]

Yes, we should be looking for the mode of the prob. distribution, and not the expectation (which is what I was doing too, though I used it only to estimate the correct value).
I should rewrite my program to actually look for the number of times we had arrived at a particular time, it should be a better estimate, though I feel it won't be far from the expectation.

Yes, it then just becomes a binning problem. I'll do that too myself, won't take a moment.

time = 1;
step = 1/6;
startdist = 0;
enddist = 30;
dist = startdist;
sumtime = 0;
iter = 100000;

times = zeros(iter, 1);

posstimes = linspace(1,24,139)-1/12;
posstimes(140) = 24+1/12;

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    
    hour = floor(time);
    minute = 10*round(floor((time-hour)*60)/10);
    times(n) = hour+minute/60;
    time = 1;
    dist = startdist;
end

averagetime = mean(times);

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

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

modetime = mode(times);

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

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

num = histc(times, posstimes);
figure; bar(posstimes, num);

So I get a mean value of around 11:50 AM and a mode of 11:20 AM. Still, even doing 100,000 iterations, I think that there may be some variance in my results as there is a bit of a shift in the histogram with repeated runs. The mode and mean are still stable though.

 

[iCyborg]

Ok, I've run 100,000,000 iterations, takes a couple of minutes though, although I could cut that by a factor of 4 since this is uber-embarassingly parallel problem and I have a quad core...

The output is:

100000000
Max times: 4834044 for time 11:20
9:50    2652411
10:10    2745505
10:20    3722470
10:30    3143163
10:40    3537828
10:50    4444865
11:0    3688679
11:10    4034842
11:20    4834044
11:30    3903873
11:40    4090896
11:50    4626295
12:0    3692413
12:10    3776528
12:20    4112389
12:30    3200844
12:40    3170546
12:50    3302839
13:0    2540418
13:10    2464346
13:20    2507556

I had the average at around 11:45, but the mode does seem to be 11:20, 100 mil. is a fairly good sample. Looks like OP's brother was wrong, as stated, 11:20 would be the answer, followed closely by 11:50...

The main function that computes the num of steps is

int NumSteps() {
	int currStep = 0;
	int currTime = 6;
	int prob = 2;
	
	while (currStep != 30) {
		double rnd = double(rand())/RAND_MAX;
		if (rnd < 1/(double(prob))) 
			currStep -= 2;
		else
			++currStep;
		if (currStep < 0) 
			currStep = 0;
		++currTime;
		
		if (currTime&#37;6 == 0) 
			++prob;
	}

	return currTime-1; 
}

which is called in a loop that bins the arrivals:

for (int i=0; i < 100000000; ++i) 
    ++FreqTable[NumSteps()];

FreqTable has size 500, and the max non-zero value I saw was 127, and the program would have crashed if some iteration took more than 500. Instead of printing out everything, the output only gives those times where the number of arrivals was at least max/2.

 

[halik]

It all really depends on what "most likely" means. I'm not sure if mode is proper thing to look at either, maybe the number of steps weighted average expected time of arrival?

Funny that you guys fired up matlab at 1am on tuesday night

 

[Lonyo]

It all really depends on what "most likely" means. I'm not sure if mode is proper thing to look at either, maybe the number of steps weighted average expected time of arrival?

Funny that you guys fired up matlab at 1am on tuesday night

Welcome to tech forums

 

[yh125d]

So, he was actually about right?

I can't help but think he didn't really get the answer himself. According to him he used only "the most basic excel formulas", and he doesn't know jack about coding and isn't particularly an excel wizard either. He's probably never heard of matlab


He just e-mailed me the .xls, I'll look at it this evening

 

[Cogman]

I think the problem is very well defined. What is the most likely time he will arrive at the sleigh?

"Arrive" means the first time he reaches his sleigh. For any time that you choose, you can calculate the probability that he first reaches his sleigh at that time. What time has the maximum probability?

The probability increase as time goes on (as the probability of taking 2 steps back decreases with time). However, it never hits 100%. When talking probability, there is no "first time he reaches his sleigh".

For example. I could say there is a 20% chance that he reaches his sleigh at 2:40. So does that mean he will most likely arrive at 2:40? I don't know, define most likely. Does that mean he COULD arrive at 2:40? Yep. You have to give a bound on what "most likely" Even though 10 hours later means he has a 1/10 chance of taking two steps back, he could still take two steps back every 10 minutes for the full hour. There isn't a high probability of that happening, but there is a probability of it happening.

 

[iCyborg]

Let's take a much simpler problem: Santa's sleigh is 1 step away.

The probability Santa will reach sleigh at:
1:10 is 0
1:20 is 0
1:30 is 0
1:40 is 0
1:50 is 0
2:00 is 1/2
2:10 is 1/4
2:20 is 1/8
2:30 is 1/16
...
3:00 is (1/64)*2/3
3:10 is (1/64)*2/9
3:20 is (1/64)*2/27
etc. etc.

I do not undestand why is it wrong to say that the most likely time Santa will reach the sleigh is at 2:00? The actual problem is much harder to compute exactly, and I don't know how to handle this resetting to 0 when he goes negative, but the problem looks well-defined.

The probability he will move forward decreases with time, but that doesn't mean the probability he arrives at the sleigh always increases, in fact, it doesn't, just like above. If it helps, let's say that the moment Santa reaches the sleigh, he takes off with it (despite risking a DUI fine...) and will no longer be making any steps. We're simply looking at what time Santa will have been at the sleigh's parking spot. Similarly, I just don't see why when talking probability, there's no "first time he reaches the sleigh"...

I could say there is a 20% chance that he reaches his sleigh at 2:40. So does that mean he will most likely arrive at 2:40? I don't know, define most likely

What's wrong with "The chance that he will reach his sleigh at any other time is <20%"?

 

[Cogman]

#include <iostream>
#include <cmath>
#include <vector>
#include <windows.h>

using namespace std;

HCRYPTPROV cHandle;

void makeCrypt()
{
    UINT ret;
    if((ret = CryptAcquireContext(&cHandle, "test", NULL, PROV_RSA_FULL, CRYPT_MACHINE_KEYSET)) == FALSE)
    {
        long long err = GetLastError();
        // Unable to grab the context, it doesn't exists
        if (err == 0x80090016L)
        {
            // Create a new context
            ret = CryptAcquireContext(&cHandle, "test", NULL, PROV_RSA_FULL, CRYPT_NEWKEYSET | CRYPT_MACHINE_KEYSET);
        }
        else
        {
            throw string("Unable to obtain cryptography context!");
        }
    }
}

int numSteps()
{
    long long total = 0;
    long long steps = 1;
    while (total < 30)
    {
        unsigned long long randNum;
        CryptGenRandom(cHandle, sizeof(unsigned long long), (BYTE*)&randNum);
        long double randPrec = randNum / pow(2.0, sizeof(unsigned long long) * 8);
        if (randPrec < (long double)1 / (long double)(steps / 6 + 1))
            total -= 2;
        else
            total += 1;
        ++steps;
    }
    return steps;
}

struct ThreadData
{
    int* activeThreads;
    int numberIterations;
    CRITICAL_SECTION* CS;
    vector<long long>* steps;
};

DWORD WINAPI MyThreadFunction( ThreadData* threadData )
{
    EnterCriticalSection(threadData->CS);
    ++(*threadData->activeThreads);
    LeaveCriticalSection(threadData->CS);
    for (long long i = 0; i < threadData->numberIterations; ++i)
    {
        long long step = numSteps();
        EnterCriticalSection(threadData->CS);
        while (step > threadData->steps->size())
            threadData->steps->push_back(0);
        (*threadData->steps)[step] += 1;
        LeaveCriticalSection(threadData->CS);
    }
    EnterCriticalSection(threadData->CS);
    --(*threadData->activeThreads);
    LeaveCriticalSection(threadData->CS);
    return 0;
}

int main()
{
    const long long MAXITER = 2097152;
    const long long NUM_THREADS = 8;
    makeCrypt();
    vector<long long> steps;
    CRITICAL_SECTION cs;
    InitializeCriticalSection(&cs);
    ThreadData threadData;
    int activeThreads = 0;
    threadData.activeThreads = &activeThreads;
    threadData.CS = &cs;
    threadData.numberIterations = MAXITER / NUM_THREADS;
    threadData.steps = &steps;

    for (int i = 0; i < NUM_THREADS; ++i)
        CreateThread(NULL, 0, (LPTHREAD_START_ROUTINE)MyThreadFunction, (void*)&threadData, 0, NULL);
    Sleep(1);
    while(activeThreads >= 1)
    {
        Sleep(1);
    }

    cout << "Probability santa will arrive AT this time\n";
    for (size_t i = 0; i < steps.size(); ++i)
    {
        int mins = i * 10;
        int hours = mins / 60;
        int days = hours / 24;
        cout << "Arrive At: " << days << " Days " << hours % 24 << " Hours " << mins % 60 << " minutes " << (long double)steps[i] / (long double)MAXITER * 100 << "%" << endl;
    }

    cout << "Probability santa will arrive BY this time\n";
    long double prob = 0;
    for (size_t i = 0; i < steps.size(); ++i)
    {
        int mins = i * 10;
        int hours = mins / 60;
        int days = hours / 24;
        prob += (long double)steps[i] / (long double)MAXITER;
        cout << days << " Days " << hours % 24 << " Hours " << mins % 60 << " minutes " << prob * 100 << "%" << endl;
    }
    DeleteCriticalSection(&cs);
    return 0;
}

Whipped this up, its a threaded C++ program using the windows API. Here are the results that I've gotten. (Oh, and it uses crytpo STRONG random numbers, not the crappy C rand() function , So I'm pretty confident in the results.)

Probability santa will arrive AT this time
Arrive At: 0 Days 0 Hours 0 minutes 0%
Arrive At: 0 Days 0 Hours 10 minutes 0%
Arrive At: 0 Days 0 Hours 20 minutes 0%
Arrive At: 0 Days 0 Hours 30 minutes 0%
Arrive At: 0 Days 0 Hours 40 minutes 0%
Arrive At: 0 Days 0 Hours 50 minutes 0%
Arrive At: 0 Days 1 Hours 0 minutes 0%
Arrive At: 0 Days 1 Hours 10 minutes 0%
Arrive At: 0 Days 1 Hours 20 minutes 0%
Arrive At: 0 Days 1 Hours 30 minutes 0%
Arrive At: 0 Days 1 Hours 40 minutes 0%
Arrive At: 0 Days 1 Hours 50 minutes 0%
Arrive At: 0 Days 2 Hours 0 minutes 0%
Arrive At: 0 Days 2 Hours 10 minutes 0%
Arrive At: 0 Days 2 Hours 20 minutes 0%
Arrive At: 0 Days 2 Hours 30 minutes 0%
Arrive At: 0 Days 2 Hours 40 minutes 0%
Arrive At: 0 Days 2 Hours 50 minutes 0%
Arrive At: 0 Days 3 Hours 0 minutes 0%
Arrive At: 0 Days 3 Hours 10 minutes 0%
Arrive At: 0 Days 3 Hours 20 minutes 0%
Arrive At: 0 Days 3 Hours 30 minutes 0%
Arrive At: 0 Days 3 Hours 40 minutes 0%
Arrive At: 0 Days 3 Hours 50 minutes 0%
Arrive At: 0 Days 4 Hours 0 minutes 0%
Arrive At: 0 Days 4 Hours 10 minutes 0%
Arrive At: 0 Days 4 Hours 20 minutes 0%
Arrive At: 0 Days 4 Hours 30 minutes 0%
Arrive At: 0 Days 4 Hours 40 minutes 0%
Arrive At: 0 Days 4 Hours 50 minutes 0%
Arrive At: 0 Days 5 Hours 0 minutes 0%
Arrive At: 0 Days 5 Hours 10 minutes 0%
Arrive At: 0 Days 5 Hours 20 minutes 0%
Arrive At: 0 Days 5 Hours 30 minutes 0%
Arrive At: 0 Days 5 Hours 40 minutes 0%
Arrive At: 0 Days 5 Hours 50 minutes 0%
Arrive At: 0 Days 6 Hours 0 minutes 0%
Arrive At: 0 Days 6 Hours 10 minutes 0%
Arrive At: 0 Days 6 Hours 20 minutes 0%
Arrive At: 0 Days 6 Hours 30 minutes 0%
Arrive At: 0 Days 6 Hours 40 minutes 0%
Arrive At: 0 Days 6 Hours 50 minutes 0%
Arrive At: 0 Days 7 Hours 0 minutes 0%
Arrive At: 0 Days 7 Hours 10 minutes 0%
Arrive At: 0 Days 7 Hours 20 minutes 0%
Arrive At: 0 Days 7 Hours 30 minutes 0%
Arrive At: 0 Days 7 Hours 40 minutes 0.000238419%
Arrive At: 0 Days 7 Hours 50 minutes 0%
Arrive At: 0 Days 8 Hours 0 minutes 0%
Arrive At: 0 Days 8 Hours 10 minutes 0.00510216%
Arrive At: 0 Days 8 Hours 20 minutes 0%
Arrive At: 0 Days 8 Hours 30 minutes 0%
Arrive At: 0 Days 8 Hours 40 minutes 0.029707%
Arrive At: 0 Days 8 Hours 50 minutes 0%
Arrive At: 0 Days 9 Hours 0 minutes 0%
Arrive At: 0 Days 9 Hours 10 minutes 0.102901%
Arrive At: 0 Days 9 Hours 20 minutes 0%
Arrive At: 0 Days 9 Hours 30 minutes 0%
Arrive At: 0 Days 9 Hours 40 minutes 0.304174%
Arrive At: 0 Days 9 Hours 50 minutes 0%
Arrive At: 0 Days 10 Hours 0 minutes 0%
Arrive At: 0 Days 10 Hours 10 minutes 0.716925%
Arrive At: 0 Days 10 Hours 20 minutes 0%
Arrive At: 0 Days 10 Hours 30 minutes 0%
Arrive At: 0 Days 10 Hours 40 minutes 1.45082%
Arrive At: 0 Days 10 Hours 50 minutes 0%
Arrive At: 0 Days 11 Hours 0 minutes 0%
Arrive At: 0 Days 11 Hours 10 minutes 2.52233%
Arrive At: 0 Days 11 Hours 20 minutes 0%
Arrive At: 0 Days 11 Hours 30 minutes 0%
Arrive At: 0 Days 11 Hours 40 minutes 3.96552%
Arrive At: 0 Days 11 Hours 50 minutes 0%
Arrive At: 0 Days 12 Hours 0 minutes 0%
Arrive At: 0 Days 12 Hours 10 minutes 5.60207%
Arrive At: 0 Days 12 Hours 20 minutes 0%
Arrive At: 0 Days 12 Hours 30 minutes 0%
Arrive At: 0 Days 12 Hours 40 minutes 7.3174%
Arrive At: 0 Days 12 Hours 50 minutes 0%
Arrive At: 0 Days 13 Hours 0 minutes 0%
Arrive At: 0 Days 13 Hours 10 minutes 8.67534%
Arrive At: 0 Days 13 Hours 20 minutes 0%
Arrive At: 0 Days 13 Hours 30 minutes 0%
Arrive At: 0 Days 13 Hours 40 minutes 9.71289%
Arrive At: 0 Days 13 Hours 50 minutes 0%
Arrive At: 0 Days 14 Hours 0 minutes 0%
Arrive At: 0 Days 14 Hours 10 minutes 10.0456%
Arrive At: 0 Days 14 Hours 20 minutes 0%
Arrive At: 0 Days 14 Hours 30 minutes 0%
Arrive At: 0 Days 14 Hours 40 minutes 9.77149%
Arrive At: 0 Days 14 Hours 50 minutes 0%
Arrive At: 0 Days 15 Hours 0 minutes 0%
Arrive At: 0 Days 15 Hours 10 minutes 8.95524%
Arrive At: 0 Days 15 Hours 20 minutes 0%
Arrive At: 0 Days 15 Hours 30 minutes 0%
Arrive At: 0 Days 15 Hours 40 minutes 7.79338%
Arrive At: 0 Days 15 Hours 50 minutes 0%
Arrive At: 0 Days 16 Hours 0 minutes 0%
Arrive At: 0 Days 16 Hours 10 minutes 6.41236%
Arrive At: 0 Days 16 Hours 20 minutes 0%
Arrive At: 0 Days 16 Hours 30 minutes 0%
Arrive At: 0 Days 16 Hours 40 minutes 5.05409%
Arrive At: 0 Days 16 Hours 50 minutes 0%
Arrive At: 0 Days 17 Hours 0 minutes 0%
Arrive At: 0 Days 17 Hours 10 minutes 3.79171%
Arrive At: 0 Days 17 Hours 20 minutes 0%
Arrive At: 0 Days 17 Hours 30 minutes 0%
Arrive At: 0 Days 17 Hours 40 minutes 2.73643%
Arrive At: 0 Days 17 Hours 50 minutes 0%
Arrive At: 0 Days 18 Hours 0 minutes 0%
Arrive At: 0 Days 18 Hours 10 minutes 1.87087%
Arrive At: 0 Days 18 Hours 20 minutes 0%
Arrive At: 0 Days 18 Hours 30 minutes 0%
Arrive At: 0 Days 18 Hours 40 minutes 1.23968%
Arrive At: 0 Days 18 Hours 50 minutes 0%
Arrive At: 0 Days 19 Hours 0 minutes 0%
Arrive At: 0 Days 19 Hours 10 minutes 0.789165%
Arrive At: 0 Days 19 Hours 20 minutes 0%
Arrive At: 0 Days 19 Hours 30 minutes 0%
Arrive At: 0 Days 19 Hours 40 minutes 0.489664%
Arrive At: 0 Days 19 Hours 50 minutes 0%
Arrive At: 0 Days 20 Hours 0 minutes 0%
Arrive At: 0 Days 20 Hours 10 minutes 0.289249%
Arrive At: 0 Days 20 Hours 20 minutes 0%
Arrive At: 0 Days 20 Hours 30 minutes 0%
Arrive At: 0 Days 20 Hours 40 minutes 0.165892%
Arrive At: 0 Days 20 Hours 50 minutes 0%
Arrive At: 0 Days 21 Hours 0 minutes 0%
Arrive At: 0 Days 21 Hours 10 minutes 0.0880241%
Arrive At: 0 Days 21 Hours 20 minutes 0%
Arrive At: 0 Days 21 Hours 30 minutes 0%
Arrive At: 0 Days 21 Hours 40 minutes 0.0504494%
Arrive At: 0 Days 21 Hours 50 minutes 0%
Arrive At: 0 Days 22 Hours 0 minutes 0%
Arrive At: 0 Days 22 Hours 10 minutes 0.0257492%
Arrive At: 0 Days 22 Hours 20 minutes 0%
Arrive At: 0 Days 22 Hours 30 minutes 0%
Arrive At: 0 Days 22 Hours 40 minutes 0.0124931%
Arrive At: 0 Days 22 Hours 50 minutes 0%
Arrive At: 0 Days 23 Hours 0 minutes 0%
Arrive At: 0 Days 23 Hours 10 minutes 0.00705719%
Arrive At: 0 Days 23 Hours 20 minutes 0%
Arrive At: 0 Days 23 Hours 30 minutes 0%
Arrive At: 0 Days 23 Hours 40 minutes 0.00286102%
Arrive At: 0 Days 23 Hours 50 minutes 0%
Arrive At: 1 Days 0 Hours 0 minutes 0%
Arrive At: 1 Days 0 Hours 10 minutes 0.00143051%
Arrive At: 1 Days 0 Hours 20 minutes 0%
Arrive At: 1 Days 0 Hours 30 minutes 0%
Arrive At: 1 Days 0 Hours 40 minutes 0.000572205%
Arrive At: 1 Days 0 Hours 50 minutes 0%
Arrive At: 1 Days 1 Hours 0 minutes 0%
Arrive At: 1 Days 1 Hours 10 minutes 0.000238419%
Arrive At: 1 Days 1 Hours 20 minutes 0%
Arrive At: 1 Days 1 Hours 30 minutes 0%
Probability santa will arrive BY this time
0 Days 0 Hours 0 minutes 0%
0 Days 0 Hours 10 minutes 0%
0 Days 0 Hours 20 minutes 0%
0 Days 0 Hours 30 minutes 0%
0 Days 0 Hours 40 minutes 0%
0 Days 0 Hours 50 minutes 0%
0 Days 1 Hours 0 minutes 0%
0 Days 1 Hours 10 minutes 0%
0 Days 1 Hours 20 minutes 0%
0 Days 1 Hours 30 minutes 0%
0 Days 1 Hours 40 minutes 0%
0 Days 1 Hours 50 minutes 0%
0 Days 2 Hours 0 minutes 0%
0 Days 2 Hours 10 minutes 0%
0 Days 2 Hours 20 minutes 0%
0 Days 2 Hours 30 minutes 0%
0 Days 2 Hours 40 minutes 0%
0 Days 2 Hours 50 minutes 0%
0 Days 3 Hours 0 minutes 0%
0 Days 3 Hours 10 minutes 0%
0 Days 3 Hours 20 minutes 0%
0 Days 3 Hours 30 minutes 0%
0 Days 3 Hours 40 minutes 0%
0 Days 3 Hours 50 minutes 0%
0 Days 4 Hours 0 minutes 0%
0 Days 4 Hours 10 minutes 0%
0 Days 4 Hours 20 minutes 0%
0 Days 4 Hours 30 minutes 0%
0 Days 4 Hours 40 minutes 0%
0 Days 4 Hours 50 minutes 0%
0 Days 5 Hours 0 minutes 0%
0 Days 5 Hours 10 minutes 0%
0 Days 5 Hours 20 minutes 0%
0 Days 5 Hours 30 minutes 0%
0 Days 5 Hours 40 minutes 0%
0 Days 5 Hours 50 minutes 0%
0 Days 6 Hours 0 minutes 0%
0 Days 6 Hours 10 minutes 0%
0 Days 6 Hours 20 minutes 0%
0 Days 6 Hours 30 minutes 0%
0 Days 6 Hours 40 minutes 0%
0 Days 6 Hours 50 minutes 0%
0 Days 7 Hours 0 minutes 0%
0 Days 7 Hours 10 minutes 0%
0 Days 7 Hours 20 minutes 0%
0 Days 7 Hours 30 minutes 0%
0 Days 7 Hours 40 minutes 0.000238419%
0 Days 7 Hours 50 minutes 0.000238419%
0 Days 8 Hours 0 minutes 0.000238419%
0 Days 8 Hours 10 minutes 0.00534058%
0 Days 8 Hours 20 minutes 0.00534058%
0 Days 8 Hours 30 minutes 0.00534058%
0 Days 8 Hours 40 minutes 0.0350475%
0 Days 8 Hours 50 minutes 0.0350475%
0 Days 9 Hours 0 minutes 0.0350475%
0 Days 9 Hours 10 minutes 0.137949%
0 Days 9 Hours 20 minutes 0.137949%
0 Days 9 Hours 30 minutes 0.137949%
0 Days 9 Hours 40 minutes 0.442123%
0 Days 9 Hours 50 minutes 0.442123%
0 Days 10 Hours 0 minutes 0.442123%
0 Days 10 Hours 10 minutes 1.15905%
0 Days 10 Hours 20 minutes 1.15905%
0 Days 10 Hours 30 minutes 1.15905%
0 Days 10 Hours 40 minutes 2.60987%
0 Days 10 Hours 50 minutes 2.60987%
0 Days 11 Hours 0 minutes 2.60987%
0 Days 11 Hours 10 minutes 5.1322%
0 Days 11 Hours 20 minutes 5.1322%
0 Days 11 Hours 30 minutes 5.1322%
0 Days 11 Hours 40 minutes 9.09772%
0 Days 11 Hours 50 minutes 9.09772%
0 Days 12 Hours 0 minutes 9.09772%
0 Days 12 Hours 10 minutes 14.6998%
0 Days 12 Hours 20 minutes 14.6998%
0 Days 12 Hours 30 minutes 14.6998%
0 Days 12 Hours 40 minutes 22.0172%
0 Days 12 Hours 50 minutes 22.0172%
0 Days 13 Hours 0 minutes 22.0172%
0 Days 13 Hours 10 minutes 30.6925%
0 Days 13 Hours 20 minutes 30.6925%
0 Days 13 Hours 30 minutes 30.6925%
0 Days 13 Hours 40 minutes 40.4054%
0 Days 13 Hours 50 minutes 40.4054%
0 Days 14 Hours 0 minutes 40.4054%
0 Days 14 Hours 10 minutes 50.451%
0 Days 14 Hours 20 minutes 50.451%
0 Days 14 Hours 30 minutes 50.451%
0 Days 14 Hours 40 minutes 60.2225%
0 Days 14 Hours 50 minutes 60.2225%
0 Days 15 Hours 0 minutes 60.2225%
0 Days 15 Hours 10 minutes 69.1777%
0 Days 15 Hours 20 minutes 69.1777%
0 Days 15 Hours 30 minutes 69.1777%
0 Days 15 Hours 40 minutes 76.9711%
0 Days 15 Hours 50 minutes 76.9711%
0 Days 16 Hours 0 minutes 76.9711%
0 Days 16 Hours 10 minutes 83.3835%
0 Days 16 Hours 20 minutes 83.3835%
0 Days 16 Hours 30 minutes 83.3835%
0 Days 16 Hours 40 minutes 88.4376%
0 Days 16 Hours 50 minutes 88.4376%
0 Days 17 Hours 0 minutes 88.4376%
0 Days 17 Hours 10 minutes 92.2293%
0 Days 17 Hours 20 minutes 92.2293%
0 Days 17 Hours 30 minutes 92.2293%
0 Days 17 Hours 40 minutes 94.9657%
0 Days 17 Hours 50 minutes 94.9657%
0 Days 18 Hours 0 minutes 94.9657%
0 Days 18 Hours 10 minutes 96.8366%
0 Days 18 Hours 20 minutes 96.8366%
0 Days 18 Hours 30 minutes 96.8366%
0 Days 18 Hours 40 minutes 98.0762%
0 Days 18 Hours 50 minutes 98.0762%
0 Days 19 Hours 0 minutes 98.0762%
0 Days 19 Hours 10 minutes 98.8654%
0 Days 19 Hours 20 minutes 98.8654%
0 Days 19 Hours 30 minutes 98.8654%
0 Days 19 Hours 40 minutes 99.3551%
0 Days 19 Hours 50 minutes 99.3551%
0 Days 20 Hours 0 minutes 99.3551%
0 Days 20 Hours 10 minutes 99.6443%
0 Days 20 Hours 20 minutes 99.6443%
0 Days 20 Hours 30 minutes 99.6443%
0 Days 20 Hours 40 minutes 99.8102%
0 Days 20 Hours 50 minutes 99.8102%
0 Days 21 Hours 0 minutes 99.8102%
0 Days 21 Hours 10 minutes 99.8982%
0 Days 21 Hours 20 minutes 99.8982%
0 Days 21 Hours 30 minutes 99.8982%
0 Days 21 Hours 40 minutes 99.9487%
0 Days 21 Hours 50 minutes 99.9487%
0 Days 22 Hours 0 minutes 99.9487%
0 Days 22 Hours 10 minutes 99.9744%
0 Days 22 Hours 20 minutes 99.9744%
0 Days 22 Hours 30 minutes 99.9744%
0 Days 22 Hours 40 minutes 99.9869%
0 Days 22 Hours 50 minutes 99.9869%
0 Days 23 Hours 0 minutes 99.9869%
0 Days 23 Hours 10 minutes 99.994%
0 Days 23 Hours 20 minutes 99.994%
0 Days 23 Hours 30 minutes 99.994%
0 Days 23 Hours 40 minutes 99.9969%
0 Days 23 Hours 50 minutes 99.9969%
1 Days 0 Hours 0 minutes 99.9969%
1 Days 0 Hours 10 minutes 99.9983%
1 Days 0 Hours 20 minutes 99.9983%
1 Days 0 Hours 30 minutes 99.9983%
1 Days 0 Hours 40 minutes 99.9989%
1 Days 0 Hours 50 minutes 99.9989%
1 Days 1 Hours 0 minutes 99.9989%
1 Days 1 Hours 10 minutes 99.9991%
1 Days 1 Hours 20 minutes 99.9991%
1 Days 1 Hours 30 minutes 99.9991%

So, if I understood the question correctly, I get that santa should arrive by 7:00pm and is most likely to arrive at 3:00pm. Which seems weird. Let me comb over the code to make sure I did things correctlyl

 

[iCyborg]

Superficially looking, you're not resetting to 0 if total goes below 0. See Born2bwire's or my function. This will make times longer.

 

[Cogman]

Superficially looking, you're not resetting to 0 if total goes below 0. See Born2bwire's or my function. This will make times longer.

Why should I reset if he wanders too far in the wrong direction? The original problem doesn't state that santa is noticed by his girlfriend and put back on the path each time he goes the wrong direction. (thus, I assume that he could stumble into the forest if he goes the wrong direction for too long)

In other words, the original problem doesn't state that santa is confined to the path between the sleigh and his gf's house.

 

[Cogman]

Let's take a much simpler problem: Santa's sleigh is 1 step away.

The probability Santa will reach sleigh at:
1:10 is 0
1:20 is 0
1:30 is 0
1:40 is 0
1:50 is 0
2:00 is 1/2
2:10 is 1/4
2:20 is 1/8
2:30 is 1/16
...
3:00 is (1/64)*2/3
3:10 is (1/64)*2/9
3:20 is (1/64)*2/27
etc. etc.

I do not undestand why is it wrong to say that the most likely time Santa will reach the sleigh is at 2:00? The actual problem is much harder to compute exactly, and I don't know how to handle this resetting to 0 when he goes negative, but the problem looks well-defined.

The probability he will move forward decreases with time, but that doesn't mean the probability he arrives at the sleigh always increases, in fact, it doesn't, just like above. If it helps, let's say that the moment Santa reaches the sleigh, he takes off with it (despite risking a DUI fine...) and will no longer be making any steps. We're simply looking at what time Santa will have been at the sleigh's parking spot. Similarly, I just don't see why when talking probability, there's no "first time he reaches the sleigh"...

What's wrong with "The chance that he will reach his sleigh at any other time is <20%"?

sorry, I was probably reading the problem wrong. I was reading it as "What time will santa most likely arrive by" rather then "What time will sant most likely arrive at". Two very different problems . I contend it is more important to know that he will arrive by a given time rather then taking bets on the exact time he will arrive.

 

[yh125d]

Why should I reset if he wanders too far in the wrong direction? The original problem doesn't state that santa is noticed by his girlfriend and put back on the path each time he goes the wrong direction. (thus, I assume that he could stumble into the forest if he goes the wrong direction for too long)

In other words, the original problem doesn't state that santa is confined to the path between the sleigh and his gf's house.

It's confined that there is only forward and backward, and he cannot step farther back than 0. I clarified this

 

[Cogman]

just a note, if I throw in the 0 reset, he arrives around 11:30. and is most likely to arrive (assuming 95&#37; is most likely) by 2:40pm

 

[Cogman]

It's confined that there is only forward and backward, and he cannot step farther back than 0. I clarified this

Ah, missed that. Ok, then my results are different then.

Probability santa will arrive AT this time
Arrive At: 0 Days 0 Hours 0 minutes 0%
Arrive At: 0 Days 0 Hours 10 minutes 0%
Arrive At: 0 Days 0 Hours 20 minutes 0%
Arrive At: 0 Days 0 Hours 30 minutes 0%
Arrive At: 0 Days 0 Hours 40 minutes 0%
Arrive At: 0 Days 0 Hours 50 minutes 0%
Arrive At: 0 Days 1 Hours 0 minutes 0%
Arrive At: 0 Days 1 Hours 10 minutes 0%
Arrive At: 0 Days 1 Hours 20 minutes 0%
Arrive At: 0 Days 1 Hours 30 minutes 0%
Arrive At: 0 Days 1 Hours 40 minutes 0%
Arrive At: 0 Days 1 Hours 50 minutes 0%
Arrive At: 0 Days 2 Hours 0 minutes 0%
Arrive At: 0 Days 2 Hours 10 minutes 0%
Arrive At: 0 Days 2 Hours 20 minutes 0%
Arrive At: 0 Days 2 Hours 30 minutes 0%
Arrive At: 0 Days 2 Hours 40 minutes 0%
Arrive At: 0 Days 2 Hours 50 minutes 0%
Arrive At: 0 Days 3 Hours 0 minutes 0%
Arrive At: 0 Days 3 Hours 10 minutes 0%
Arrive At: 0 Days 3 Hours 20 minutes 0%
Arrive At: 0 Days 3 Hours 30 minutes 0%
Arrive At: 0 Days 3 Hours 40 minutes 0%
Arrive At: 0 Days 3 Hours 50 minutes 0%
Arrive At: 0 Days 4 Hours 0 minutes 0%
Arrive At: 0 Days 4 Hours 10 minutes 0%
Arrive At: 0 Days 4 Hours 20 minutes 0%
Arrive At: 0 Days 4 Hours 30 minutes 0%
Arrive At: 0 Days 4 Hours 40 minutes 0%
Arrive At: 0 Days 4 Hours 50 minutes 0%
Arrive At: 0 Days 5 Hours 0 minutes 0%
Arrive At: 0 Days 5 Hours 10 minutes 0%
Arrive At: 0 Days 5 Hours 20 minutes 0%
Arrive At: 0 Days 5 Hours 30 minutes 0%
Arrive At: 0 Days 5 Hours 40 minutes 0%
Arrive At: 0 Days 5 Hours 50 minutes 0%
Arrive At: 0 Days 6 Hours 0 minutes 0.00201464%
Arrive At: 0 Days 6 Hours 10 minutes 0.00167489%
Arrive At: 0 Days 6 Hours 20 minutes 0.0030458%
Arrive At: 0 Days 6 Hours 30 minutes 0.0205755%
Arrive At: 0 Days 6 Hours 40 minutes 0.0195861%
Arrive At: 0 Days 6 Hours 50 minutes 0.0319242%
Arrive At: 0 Days 7 Hours 0 minutes 0.0995576%
Arrive At: 0 Days 7 Hours 10 minutes 0.0977218%
Arrive At: 0 Days 7 Hours 20 minutes 0.149423%
Arrive At: 0 Days 7 Hours 30 minutes 0.339347%
Arrive At: 0 Days 7 Hours 40 minutes 0.318813%
Arrive At: 0 Days 7 Hours 50 minutes 0.450176%
Arrive At: 0 Days 8 Hours 0 minutes 0.833857%
Arrive At: 0 Days 8 Hours 10 minutes 0.769615%
Arrive At: 0 Days 8 Hours 20 minutes 1.01339%
Arrive At: 0 Days 8 Hours 30 minutes 1.6655%
Arrive At: 0 Days 8 Hours 40 minutes 1.47479%
Arrive At: 0 Days 8 Hours 50 minutes 1.81471%
Arrive At: 0 Days 9 Hours 0 minutes 2.65657%
Arrive At: 0 Days 9 Hours 10 minutes 2.31749%
Arrive At: 0 Days 9 Hours 20 minutes 2.74377%
Arrive At: 0 Days 9 Hours 30 minutes 3.7203%
Arrive At: 0 Days 9 Hours 40 minutes 3.14285%
Arrive At: 0 Days 9 Hours 50 minutes 3.5403%
Arrive At: 0 Days 10 Hours 0 minutes 4.44208%
Arrive At: 0 Days 10 Hours 10 minutes 3.69173%
Arrive At: 0 Days 10 Hours 20 minutes 4.04403%
Arrive At: 0 Days 10 Hours 30 minutes 4.83111%
Arrive At: 0 Days 10 Hours 40 minutes 3.90332%
Arrive At: 0 Days 10 Hours 50 minutes 4.08552%
Arrive At: 0 Days 11 Hours 0 minutes 4.61689%
Arrive At: 0 Days 11 Hours 10 minutes 3.68591%
Arrive At: 0 Days 11 Hours 20 minutes 3.78062%
Arrive At: 0 Days 11 Hours 30 minutes 4.11302%
Arrive At: 0 Days 11 Hours 40 minutes 3.20497%
Arrive At: 0 Days 11 Hours 50 minutes 3.17214%
Arrive At: 0 Days 12 Hours 0 minutes 3.29905%
Arrive At: 0 Days 12 Hours 10 minutes 2.54677%
Arrive At: 0 Days 12 Hours 20 minutes 2.46679%
Arrive At: 0 Days 12 Hours 30 minutes 2.50294%
Arrive At: 0 Days 12 Hours 40 minutes 1.89126%
Arrive At: 0 Days 12 Hours 50 minutes 1.77518%
Arrive At: 0 Days 13 Hours 0 minutes 1.738%
Arrive At: 0 Days 13 Hours 10 minutes 1.30007%
Arrive At: 0 Days 13 Hours 20 minutes 1.2027%
Arrive At: 0 Days 13 Hours 30 minutes 1.15272%
Arrive At: 0 Days 13 Hours 40 minutes 0.843364%
Arrive At: 0 Days 13 Hours 50 minutes 0.761306%
Arrive At: 0 Days 14 Hours 0 minutes 0.709814%
Arrive At: 0 Days 14 Hours 10 minutes 0.51741%
Arrive At: 0 Days 14 Hours 20 minutes 0.456059%
Arrive At: 0 Days 14 Hours 30 minutes 0.41759%
Arrive At: 0 Days 14 Hours 40 minutes 0.297374%
Arrive At: 0 Days 14 Hours 50 minutes 0.256014%
Arrive At: 0 Days 15 Hours 0 minutes 0.229079%
Arrive At: 0 Days 15 Hours 10 minutes 0.163376%
Arrive At: 0 Days 15 Hours 20 minutes 0.139147%
Arrive At: 0 Days 15 Hours 30 minutes 0.121343%
Arrive At: 0 Days 15 Hours 40 minutes 0.0852048%
Arrive At: 0 Days 15 Hours 50 minutes 0.0718653%
Arrive At: 0 Days 16 Hours 0 minutes 0.0605643%
Arrive At: 0 Days 16 Hours 10 minutes 0.042218%
Arrive At: 0 Days 16 Hours 20 minutes 0.0348687%
Arrive At: 0 Days 16 Hours 30 minutes 0.0298083%
Arrive At: 0 Days 16 Hours 40 minutes 0.0205994%
Arrive At: 0 Days 16 Hours 50 minutes 0.0168622%
Arrive At: 0 Days 17 Hours 0 minutes 0.0132024%
Arrive At: 0 Days 17 Hours 10 minutes 0.00936389%
Arrive At: 0 Days 17 Hours 20 minutes 0.007236%
Arrive At: 0 Days 17 Hours 30 minutes 0.00606775%
Arrive At: 0 Days 17 Hours 40 minutes 0.00431538%
Arrive At: 0 Days 17 Hours 50 minutes 0.00319481%
Arrive At: 0 Days 18 Hours 0 minutes 0.00248551%
Arrive At: 0 Days 18 Hours 10 minutes 0.00166297%
Arrive At: 0 Days 18 Hours 20 minutes 0.00128746%
Arrive At: 0 Days 18 Hours 30 minutes 0.00104904%
Arrive At: 0 Days 18 Hours 40 minutes 0.000703335%
Arrive At: 0 Days 18 Hours 50 minutes 0.000447035%
Arrive At: 0 Days 19 Hours 0 minutes 0.000399351%
Arrive At: 0 Days 19 Hours 10 minutes 0.000232458%
Arrive At: 0 Days 19 Hours 20 minutes 0.000166893%
Arrive At: 0 Days 19 Hours 30 minutes 0.000166893%
Arrive At: 0 Days 19 Hours 40 minutes 7.15256e-005%
Arrive At: 0 Days 19 Hours 50 minutes 7.15256e-005%
Arrive At: 0 Days 20 Hours 0 minutes 5.36442e-005%
Arrive At: 0 Days 20 Hours 10 minutes 3.57628e-005%
Arrive At: 0 Days 20 Hours 20 minutes 2.38419e-005%
Arrive At: 0 Days 20 Hours 30 minutes 0%
Arrive At: 0 Days 20 Hours 40 minutes 5.96046e-006%
Arrive At: 0 Days 20 Hours 50 minutes 0%
Arrive At: 0 Days 21 Hours 0 minutes 1.78814e-005%
Arrive At: 0 Days 21 Hours 10 minutes 5.96046e-006%
Arrive At: 0 Days 21 Hours 20 minutes 0%
Probability santa will arrive BY this time
0 Days 0 Hours 0 minutes 0%
0 Days 0 Hours 10 minutes 0%
0 Days 0 Hours 20 minutes 0%
0 Days 0 Hours 30 minutes 0%
0 Days 0 Hours 40 minutes 0%
0 Days 0 Hours 50 minutes 0%
0 Days 1 Hours 0 minutes 0%
0 Days 1 Hours 10 minutes 0%
0 Days 1 Hours 20 minutes 0%
0 Days 1 Hours 30 minutes 0%
0 Days 1 Hours 40 minutes 0%
0 Days 1 Hours 50 minutes 0%
0 Days 2 Hours 0 minutes 0%
0 Days 2 Hours 10 minutes 0%
0 Days 2 Hours 20 minutes 0%
0 Days 2 Hours 30 minutes 0%
0 Days 2 Hours 40 minutes 0%
0 Days 2 Hours 50 minutes 0%
0 Days 3 Hours 0 minutes 0%
0 Days 3 Hours 10 minutes 0%
0 Days 3 Hours 20 minutes 0%
0 Days 3 Hours 30 minutes 0%
0 Days 3 Hours 40 minutes 0%
0 Days 3 Hours 50 minutes 0%
0 Days 4 Hours 0 minutes 0%
0 Days 4 Hours 10 minutes 0%
0 Days 4 Hours 20 minutes 0%
0 Days 4 Hours 30 minutes 0%
0 Days 4 Hours 40 minutes 0%
0 Days 4 Hours 50 minutes 0%
0 Days 5 Hours 0 minutes 0%
0 Days 5 Hours 10 minutes 0%
0 Days 5 Hours 20 minutes 0%
0 Days 5 Hours 30 minutes 0%
0 Days 5 Hours 40 minutes 0%
0 Days 5 Hours 50 minutes 0%
0 Days 6 Hours 0 minutes 0.00201464%
0 Days 6 Hours 10 minutes 0.00368953%
0 Days 6 Hours 20 minutes 0.00673532%
0 Days 6 Hours 30 minutes 0.0273108%
0 Days 6 Hours 40 minutes 0.0468969%
0 Days 6 Hours 50 minutes 0.0788212%
0 Days 7 Hours 0 minutes 0.178379%
0 Days 7 Hours 10 minutes 0.276101%
0 Days 7 Hours 20 minutes 0.425524%
0 Days 7 Hours 30 minutes 0.764871%
0 Days 7 Hours 40 minutes 1.08368%
0 Days 7 Hours 50 minutes 1.53386%
0 Days 8 Hours 0 minutes 2.36772%
0 Days 8 Hours 10 minutes 3.13733%
0 Days 8 Hours 20 minutes 4.15072%
0 Days 8 Hours 30 minutes 5.81622%
0 Days 8 Hours 40 minutes 7.291%
0 Days 8 Hours 50 minutes 9.10571%
0 Days 9 Hours 0 minutes 11.7623%
0 Days 9 Hours 10 minutes 14.0798%
0 Days 9 Hours 20 minutes 16.8235%
0 Days 9 Hours 30 minutes 20.5438%
0 Days 9 Hours 40 minutes 23.6867%
0 Days 9 Hours 50 minutes 27.227%
0 Days 10 Hours 0 minutes 31.669%
0 Days 10 Hours 10 minutes 35.3608%
0 Days 10 Hours 20 minutes 39.4048%
0 Days 10 Hours 30 minutes 44.2359%
0 Days 10 Hours 40 minutes 48.1392%
0 Days 10 Hours 50 minutes 52.2247%
0 Days 11 Hours 0 minutes 56.8416%
0 Days 11 Hours 10 minutes 60.5276%
0 Days 11 Hours 20 minutes 64.3082%
0 Days 11 Hours 30 minutes 68.4212%
0 Days 11 Hours 40 minutes 71.6262%
0 Days 11 Hours 50 minutes 74.7983%
0 Days 12 Hours 0 minutes 78.0973%
0 Days 12 Hours 10 minutes 80.6441%
0 Days 12 Hours 20 minutes 83.1109%
0 Days 12 Hours 30 minutes 85.6138%
0 Days 12 Hours 40 minutes 87.5051%
0 Days 12 Hours 50 minutes 89.2803%
0 Days 13 Hours 0 minutes 91.0183%
0 Days 13 Hours 10 minutes 92.3184%
0 Days 13 Hours 20 minutes 93.521%
0 Days 13 Hours 30 minutes 94.6738%
0 Days 13 Hours 40 minutes 95.5171%
0 Days 13 Hours 50 minutes 96.2784%
0 Days 14 Hours 0 minutes 96.9882%
0 Days 14 Hours 10 minutes 97.5057%
0 Days 14 Hours 20 minutes 97.9617%
0 Days 14 Hours 30 minutes 98.3793%
0 Days 14 Hours 40 minutes 98.6767%
0 Days 14 Hours 50 minutes 98.9327%
0 Days 15 Hours 0 minutes 99.1618%
0 Days 15 Hours 10 minutes 99.3252%
0 Days 15 Hours 20 minutes 99.4643%
0 Days 15 Hours 30 minutes 99.5856%
0 Days 15 Hours 40 minutes 99.6708%
0 Days 15 Hours 50 minutes 99.7427%
0 Days 16 Hours 0 minutes 99.8033%
0 Days 16 Hours 10 minutes 99.8455%
0 Days 16 Hours 20 minutes 99.8804%
0 Days 16 Hours 30 minutes 99.9102%
0 Days 16 Hours 40 minutes 99.9308%
0 Days 16 Hours 50 minutes 99.9476%
0 Days 17 Hours 0 minutes 99.9608%
0 Days 17 Hours 10 minutes 99.9702%
0 Days 17 Hours 20 minutes 99.9774%
0 Days 17 Hours 30 minutes 99.9835%
0 Days 17 Hours 40 minutes 99.9878%
0 Days 17 Hours 50 minutes 99.991%
0 Days 18 Hours 0 minutes 99.9935%
0 Days 18 Hours 10 minutes 99.9952%
0 Days 18 Hours 20 minutes 99.9964%
0 Days 18 Hours 30 minutes 99.9975%
0 Days 18 Hours 40 minutes 99.9982%
0 Days 18 Hours 50 minutes 99.9986%
0 Days 19 Hours 0 minutes 99.999%
0 Days 19 Hours 10 minutes 99.9993%
0 Days 19 Hours 20 minutes 99.9994%
0 Days 19 Hours 30 minutes 99.9996%
0 Days 19 Hours 40 minutes 99.9997%
0 Days 19 Hours 50 minutes 99.9998%
0 Days 20 Hours 0 minutes 99.9998%
0 Days 20 Hours 10 minutes 99.9998%
0 Days 20 Hours 20 minutes 99.9999%
0 Days 20 Hours 30 minutes 99.9999%
0 Days 20 Hours 40 minutes 99.9999%
0 Days 20 Hours 50 minutes 99.9999%
0 Days 21 Hours 0 minutes 99.9999%
0 Days 21 Hours 10 minutes 99.9999%
0 Days 21 Hours 20 minutes 99.9999%

New results for any that care . In other words. .95 probability that he will arrive by 2:40pm with 11:30am being the most likely arrival time. This was done with 16 million iterations.

 

[DrPizza]

wtf? How are any of you answering this without knowing if 2:30 is rounded to 2, or rounded to 3?

 

[yh125d]

wtf? How are any of you answering this without knowing if 2:30 is rounded to 2, or rounded to 3?

The question doesn't ask which hour

 

[Kyteland]

I get 11:50 as the answer using some basic assumptions and some simple excel work.

At 1:00 there is 100&#37; probability he is at step 0, 0% probability he is at any other step.

Step 0 transitions: backwards from steps 0,1,2 with probability 1/T

Step 1-27 transitions generalized to N: backwards from step N+2 with probability 1/T, forward from step N-1 with probability 1-1/T

Step 28 transitions: forward from step 27 with probability 1-1/T
Step 29 transitions: forward from step 28 with probability 1-1/T
Step 30 transitions: forward from step 29 with probability 1-1/T + probability of already being at step 30.

At 11:40 the probability of being at step 30 is 48.1419%
At 11:50 the probability of being at step 30 is 52.2296%

 

[Cogman]

wtf? How are any of you answering this without knowing if 2:30 is rounded to 2, or rounded to 3?

T = the current hour, rounded down

2:59:99 is 3 for the probability of steps taken backwards. That's the only place rounding ocures.

I still like my solution the best, just because it uses a CSPRNG. Making it slow, but precise. (and its threaded )

 

[iCyborg]

I still like my solution the best, just because it uses a CSPRNG. Making it slow, but precise. (and its threaded )

I think it's not necessary to be unpredictable for statistics purposes as long as the distribution is uniform over [0,1], and CryptGenRandom is only better because its seed is "more" random.
Also, I would have used the return values from CreateThread calls to populate an array of handles, and then use WaitForMultipleObjects with wait-all to wait for all of them to finish, instead of this activeThreads variable which is checked every 1s.

Anyway, both of this is nitpicking. But something seems fishy. Here are your results from 10:10 to 11:10

Arrive At: 0 Days 10 Hours 10 minutes 3.69173%
Arrive At: 0 Days 10 Hours 20 minutes 4.04403%
Arrive At: 0 Days 10 Hours 30 minutes 4.83111%
Arrive At: 0 Days 10 Hours 40 minutes 3.90332%
Arrive At: 0 Days 10 Hours 50 minutes 4.08552%
Arrive At: 0 Days 11 Hours 0 minutes 4.61689%
Arrive At: 0 Days 11 Hours 10 minutes 3.68591%

(You said the most likely time is 11:00, but it looks like it's 10:30)

And here are mine, but shifted by 50mins into the future (I converted into percentage like yours):

11:0 - 3.688679%
11:10 - 4.034842%
11:20 - 4.834044%
11:30 - 3.903873%
11:40 - 4.090896%
11:50 - 4.626295%
12:0 - 3.692413%

They match remarkably, except that they're off by 50mins. Not sure who is wrong, but since Born2bwire has the same final results in Matlab as I do, I'm gonna go with the "Argumentum ad populum"

 

[Cogman]

I think it's not necessary to be unpredictable for statistics purposes as long as the distribution is uniform over [0,1], and CryptGenRandom is only better because its seed is "more" random.
Also, I would have used the return values from CreateThread calls to populate an array of handles, and then use WaitForMultipleObjects with wait-all to wait for all of them to finish, instead of this activeThreads variable which is checked every 1s.

Anyway, both of this is nitpicking. But something seems fishy. Here are your results from 10:10 to 11:10

Arrive At: 0 Days 10 Hours 10 minutes 3.69173&#37;
Arrive At: 0 Days 10 Hours 20 minutes 4.04403%
Arrive At: 0 Days 10 Hours 30 minutes 4.83111%
Arrive At: 0 Days 10 Hours 40 minutes 3.90332%
Arrive At: 0 Days 10 Hours 50 minutes 4.08552%
Arrive At: 0 Days 11 Hours 0 minutes 4.61689%
Arrive At: 0 Days 11 Hours 10 minutes 3.68591%

(You said the most likely time is 11:00, but it looks like it's 10:30)

And here are mine, but shifted by 50mins into the future (I converted into percentage like yours):

11:0 - 3.688679%
11:10 - 4.034842%
11:20 - 4.834044%
11:30 - 3.903873%
11:40 - 4.090896%
11:50 - 4.626295%
12:0 - 3.692413%

They match remarkably, except that they're off by 50mins. Not sure who is wrong, but since Born2bwire has the same final results in Matlab as I do, I'm gonna go with the "Argumentum ad populum"

It might be off by one. Mine is the hours from 1am, yours is the actual time. We have essentially the same answer except mine might be off by one.

What I find remarkable is the alternating nature of the results. It must have to do with the two steps back thing.

 

[mjrpes3]

The probability of number steps at each hour is 6*(x-3)/x, which is simplified from probability step back as -2/x and probability step forward as (x-1)/x, times 6 (the number of step attempts per hour). Ignore time < 4 because probability is negative.

Plug that formula into Excel and you get:

Time / Steps / Sum
04 1.50 1.50
05 2.40 3.90
06 3.00 6.90
07 3.42 10.32
08 3.75 14.07
09 4.00 18.07
10 4.20 22.27
11 4.36 26.64
12 4.50 31.14
13 4.61 35.75
14 4.71 40.47

So probability is that he will reach 30 steps near the end of 11:00am.

 

[DrPizza]

2:59:99 is 3 for the probability of steps taken backwards. That's the only place rounding ocures.

I still like my solution the best, just because it uses a CSPRNG. Making it slow, but precise. (and its threaded )

Oh duhhhhh. I only read the OP once. Somehow, I missed the word "down."