The Monty Hall Problem

[theonefoster]

I'm writing a quick a simulation for The Monty Hall Problem but it comes up with winning a car 75% of the time, even with 20k simulations. It does that every time, but I can't understand why. It should win a car 66% of the time. Any ideas? Here's my procedure for a single simulation:

Private Function Simulate() As Boolean 'true means a car was won; false means a goat was won
        Dim Door(2) As Boolean
        Dim doornum As SByte
        Dim choice As SByte = 0 'initially choose the first door every time

        Randomize()
        doornum = Math.Round(Rnd() * 2) 'generate a random number between 0 and 2
        Door(doornum) = True 'set a random door to have a car behind it

        If Door(1) = False Then 'if the second (zero-based) door is opened,
            choice = 2 'choose the third
        ElseIf Door(2) = False Then 'if the third door is opened,
            choice = 1 'choose the second
        End If 'change choice to one of the other doors

        If Door(choice) Then 'if the new choice has a car,
            Return True
        Else 'if the new choice doesn't have a car
            Return False
        End If

    End Function

Cheers

[techgnome]

Nope... 75% of the time is correct... Mythbusters covered this once on why it comes out the way it does....
Mythbusters Monty Hall

-tg

[Inferrd]

I don't know what answer Myth Busters came up with as all the video links seem to have been taken down by the copyright police. However, if it's the standard Monty Hall problem then the answer should be 66%.

The reason you are getting 75% is that your method for generating random numbers is somewhat skewed (in this particular case you'll get twice as many 1's as 0's or 2's).

There's a couple of easy fixes:

1) If using the VBMath.Rnd() method, use the formula recommended in the MSDN here which in this case would simplify to

doornum = CInt(Math.Floor(3 * Rnd()))

2) Create an instance of the Random Class. You would create only one instance at Class level (prevents sequence repetition) and use its .Next method in your function.

Public Class Whatever
    Private rnGen As New Random ' at Class Level

    Private Function Simulate() As Boolean 'true means a car was won; false means a goat was won
        Dim Door(2) As Boolean
        Dim doornum As SByte
        Dim choice As SByte = 0 'initially choose the first door every time

           doornum = rnGen.Next(3)

        ' code

    End Function

End Class

[theonefoster]

Thanks very much Inferrd, that was exactly the problem. I replaced Math.Round(Rnd() * 2) with Int(Rnd() * 3) and it worked great. I do understand where I went wrong now - half of the real numbers between 0 and 2 round to 1, but only a quarter round to each 0 and 2.

Thanks again

[Shaggy Hiker]

Since this is .NET, you shouldn't be using Randomize and Rnd. You should scrap that in favor of the Random object. For one thing, the way you are using Randomize will cause you trouble if you were to test the program by calling Simulate in a tight loop. The issue is that Randomize will seed the random number generator with the current system time down to the second. By calling Simulate in a tight loop, you would end up calling Randomize more than once per second, which would mean that it would seed more than once with the same value, which would mean that your sequence would be exactly the same over and over and over, until the second advanced and you got a new sequence again and again.

The Random object has the same issue, of course, but if you create it a single Random object at form scope then you won't have that issue. You could also solve the problem by calling Randomize only one time, such as in Form Load, but what's the point of using Rnd, which sucks, versus the GetNext method of Random, which is far easier to get right?