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   Re: Probability Problem 1
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   Author  Topic: Re: Probability Problem 1  (Read 81 times)
towr
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Re: Probability Problem 1  
« on: Jun 17th, 2018, 12:14pm »
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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
 
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towr
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Re: Probability Problem 1  
« Reply #1 on: Jun 17th, 2018, 10:52pm »
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Ah 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%
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Re: Probability Problem 1  
« Reply #2 on: Jun 18th, 2018, 4:36pm »
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A quick and dirty program to count all possibilities agrees with towr's second answer.
 
C# code:
hidden:

   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();
   }

Output: 23184 successes out of 65536 total
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Re: Probability Problem 1  
« Reply #3 on: Jun 18th, 2018, 8:59pm »
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Your second answer is correct towr.
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