An interview question: About Probability

Question

An interview question:

Given a function f(x) that 1/4 times returns 0, 3/4 times returns 1. Write a function g(x) using f(x) that 1/2 times returns 0, 1/2 times returns

Answer 1

This is much like the Monty Hall paradox.

In general.

Public Class Form1

    'the general case
    '
    'twiceThis = 2 is 1 in four chance of 0
    'twiceThis = 3 is 1 in six chance of 0
    '
    'twiceThis = x is 1 in 2x chance of 0

    Const twiceThis As Integer = 7
    Const numOf As Integer = twiceThis * 2

    Private Sub Button1_Click(ByVal sender As System.Object, _
                              ByVal e As System.EventArgs) Handles Button1.Click

        Const tries As Integer = 1000
        y = New List(Of Integer)

        Dim ct0 As Integer = 0
        Dim ct1 As Integer = 0
        Debug.WriteLine("")
        ''show all possible values of fx
        'For x As Integer = 1 To numOf
        '    Debug.WriteLine(fx)
        'Next

        'test that gx returns 50% 0's and 50% 1's
        Dim stpw As New Stopwatch
        stpw.Start()
        For x As Integer = 1 To tries
            Dim g_x As Integer = gx()
            'Debug.WriteLine(g_x.ToString) 'used to verify that gx returns 0 or 1 randomly
            If g_x = 0 Then ct0 += 1 Else ct1 += 1
        Next
        stpw.Stop()
        'the results
        Debug.WriteLine((ct0 / tries).ToString("p1"))
        Debug.WriteLine((ct1 / tries).ToString("p1"))
        Debug.WriteLine((stpw.ElapsedTicks / tries).ToString("n0"))

    End Sub

    Dim prng As New Random
    Dim y As New List(Of Integer)

    Private Function fx() As Integer

        '1 in numOf chance of zero being returned
        If y.Count = 0 Then
            'reload y
            y.Add(0) 'fx has only one zero value
            Do
                y.Add(1) 'the rest are ones
            Loop While y.Count < numOf
        End If
        'return a random value 
        Dim idx As Integer = prng.Next(y.Count)
        Dim rv As Integer = y(idx)
        y.RemoveAt(idx) 'remove the value selected
        Return rv

    End Function

    Private Function gx() As Integer

        'a function g(x) using f(x) that 50% of the time returns 0
        '                           that 50% of the time returns 1
        Dim rv As Integer = 0
        For x As Integer = 1 To twiceThis
            fx()
        Next
        For x As Integer = 1 To twiceThis
            rv += fx()
        Next
        If rv = twiceThis Then Return 1 Else Return 0

    End Function
End Class