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        riddles >> hard >> Re: Probability Problem 1
        
(Message started by: towr on Jun 17^th, 2018, 12:14pm)

Title: Re: Probability Problem 1
Post by towr on Jun 17^th, 2018, 12:14pm

[hide]
experimentally it's around 36%

mathematically, I think it should be 4 times the number of ways to split 8 passengers over 3 non-empty carriages, divided by the number of ways to split 8 passengers over 4 carriages.
Which I'd hope would be 4*([5+3-1]!/5!/[3-1]!) / ([8+4-1]!/8!/[4-1]!), but isn't. So I'm making a mistake somewhere

[/hide]

Title: Re: Probability Problem 1
Post by towr on Jun 17^th, 2018, 10:52pm

Ah [hide]like railway companies, I had trouble looking at the passengers as individuals.

It should be (4*3^8 - 12*2^8 + 12) / 4**8 = 1449/4096 ~= 35.4%
[/hide]

Title: Re: Probability Problem 1
Post by rmsgrey on Jun 18^th, 2018, 4:36pm

A quick and dirty program to count all possibilities agrees with towr's second answer.

C# code:
[hideb]
static void Main(string[] args)
{
int[] pass = {0,0,0,0,0,0,0,0};
int total = 0;
int success = 0;
while (pass[7] < 4)
{
int[] car = { 0, 0, 0, 0 };
for (int i = 0; i < 8; i++)
car[pass[i] = 1;
if (car[0] + car[1] + car[2] + car[3] == 3)
success++;
total++;
int j = 0;
pass[j]++;
while (j < 7 && pass[j] == 4)
{
pass[j] = 0;
j++;
pass[j]++;
}
}
Console.WriteLine("{0} successes out of {1} total", success, total);
Console.ReadLine();
}
[/hideb]
Output: [hide]23184 successes out of 65536 total[/hide]

Title: Re: Probability Problem 1
Post by navdeep1771 on Jun 18^th, 2018, 8:59pm

Your second answer is correct towr.