delphidelphi-xe2
Random number with equal total
I have 4 number
a,b,c,d : integers
i need to have a random number between 2-7 assigned to each one but the total of all four numbers has to be 22
how can i do this?
Solution
First of all I'm going to make it clear that as stated the question does not uniquely define the problem. You ask for random sampling, but do not specify the desired distribution of the samples.
It's a common abuse of mathematical terminology to say random when you actually mean uniformly distributed. So I'm going to assume that's what you mean. Specifically that you want all possible distinct sets of 4 numbers to have equal probability of selection. The simplest and most efficient way to achieve this is as follows:
- Enumerate all such possible sets of 4 numbers.
- Count these sets of numbers, say N.
- To sample, choose random number, i say, from uniform distribution in range 0 to N-1.
- Return the i-th set of 4 numbers.
The list of possible distinct sets is small. Off the top of my head I'd guess there are around 50 candidates.
Generating the list of candidates is quite simple. Just run three nested for loops from 2 to 7. This gives you combinations of the first three numbers. Add them up, subtract from 22 and check if the final number is in range.
Since you seem to like to see code, here's a simple demonstration:
{$APPTYPE CONSOLE}
uses
System.Math,
Generics.Collections;
type
TValue = record
a, b, c, d: Integer;
procedure Write;
end;
procedure TValue.Write;
begin
Writeln(a, ' ', b, ' ', c, ' ', d);
end;
var
Combinations: TArray<TValue>;
procedure InitialiseCombinations;
var
a, b, c, d: Integer;
Value: TValue;
List: TList<TValue>;
begin
List := TList<TValue>.Create;
try
for a := 2 to 7 do
for b := 2 to 7 do
for c := 2 to 7 do
begin
d := 22 - a - b - c;
if InRange(d, 2, 7) then
begin
Value.a := a;
Value.b := b;
Value.c := c;
Value.d := d;
List.Add(Value);
end;
end;
Combinations := List.ToArray;
finally
List.Free;
end;
end;
function GetSample: TValue;
begin
Result := Combinations[Random(Length(Combinations))];
end;
var
i: Integer;
begin
Randomize;
InitialiseCombinations;
for i := 1 to 25 do
GetSample.Write;
Readln;
end.
It's clear from inspection that this algorithm samples from the available values uniformly.
But what about the other proposed algorithms. We can perform a crude heuristic test by sampling repeatedly and counting how many times each possible sample is produced. Here it is:
{$APPTYPE CONSOLE}
uses
System.SysUtils,
System.Math,
Generics.Collections;
type
TValue = record
a, b, c, d: Integer;
procedure Write;
class operator Equal(const lhs, rhs: TValue): Boolean;
end;
procedure TValue.Write;
begin
Writeln(a, ' ', b, ' ', c, ' ', d);
end;
class operator TValue.Equal(const lhs, rhs: TValue): Boolean;
begin
Result := (lhs.a=rhs.a) and (lhs.b=rhs.b) and (lhs.c=rhs.c) and (lhs.d=rhs.d);
end;
var
Combinations: TArray<TValue>;
procedure InitialiseCombinations;
var
a, b, c, d: Integer;
Value: TValue;
List: TList<TValue>;
begin
List := TList<TValue>.Create;
try
for a := 2 to 7 do
for b := 2 to 7 do
for c := 2 to 7 do
begin
d := 22 - a - b - c;
if InRange(d, 2, 7) then
begin
Value.a := a;
Value.b := b;
Value.c := c;
Value.d := d;
List.Add(Value);
end;
end;
Combinations := List.ToArray;
finally
List.Free;
end;
end;
function GetSampleHeffernan: TValue;
begin
Result := Combinations[Random(Length(Combinations))];
end;
function GetSampleVanDien: TValue;
const
TOTAL = 22;
VALUE_COUNT = 4;
MIN_VALUE = 2;
MAX_VALUE = 7;
var
Values: array[0..VALUE_COUNT-1] of Integer;
Shortage: Integer;
Candidates: TList<Integer>;
ValueIndex: Integer;
CandidateIndex: Integer;
begin
Assert(VALUE_COUNT * MAX_VALUE >= TOTAL, 'Total can never be reached!');
Assert(VALUE_COUNT * MIN_VALUE <= TOTAL, 'Total is always exceeded!');
Randomize;
Candidates := TList<Integer>.Create;
try
for ValueIndex := 0 to VALUE_COUNT-1 do
begin
Values[ValueIndex] := MIN_VALUE;
Candidates.Add(ValueIndex);
end;
Shortage := TOTAL - VALUE_COUNT * MIN_VALUE;
while Shortage > 0 do
begin
CandidateIndex := Random(Candidates.Count);
ValueIndex := Candidates[CandidateIndex];
Values[ValueIndex] := Values[ValueIndex] + 1;
if Values[ValueIndex] = MAX_VALUE then
Candidates.Remove(CandidateIndex);
Shortage := Shortage - 1;
end;
finally
Candidates.Free;
end;
Result.a := Values[0];
Result.b := Values[1];
Result.c := Values[2];
Result.d := Values[3];
end;
function GetSampleLama: TValue;
type
TRandomValues = array[1..4] of Integer;
var
IntSum: Integer;
Values: TRandomValues;
begin
// initialize a helper variable for calculating sum of the generated numbers
IntSum := 0;
// in the first step just generate a number in the range of 2 to 7 and store
// it to the first integer element
Values[1] := RandomRange(2, 7);
// and increment the sum value
IntSum := IntSum + Values[1];
// as the next step we need to generate number, but here we need also say in
// which range by the following rules to ensure we ever reach 22 (consider, if
// the 1st number was e.g. 3, then you can't generate the second number smaller
// than 5 because then even if the next two numbers would be max, you would get
// e.g. only 3 + 4 + 7 + 7 = 21, so just use this rule:
// Values[1] Values[2]
// 2 6..7
// 3 5..7
// 4 4..7
// 5 3..7
// 6..7 2..7
Values[2] := RandomRange(Max(2, 8 - Values[1]), 7);
// and increment the sum value
IntSum := IntSum + Values[2];
// if the third step we need to generate a value in the range of 15 to 20 since
// the fourth number can be still in the range of 2 to 7 which means that the sum
// after this step must be from 22-7 to 22-2 which is 15 to 20, so let's generate
// a number which will fit into this sum
Values[3] := RandomRange(Max(2, Min(7, 15 - IntSum)), Max(2, Min(7, 20 - IntSum)));
// and for the last number let's just take 22 and subtract the sum of all previous
// numbers
Values[4] := 22 - (IntSum + Values[3]);
Result.a := Values[1];
Result.b := Values[2];
Result.c := Values[3];
Result.d := Values[4];
end;
function IndexOf(const Value: TValue): Integer;
begin
for Result := 0 to high(Combinations) do
if Combinations[Result] = Value then
exit;
raise EAssertionFailed.Create('Invalid value');
end;
procedure CheckCounts(const Name: string; const GetSample: TFunc<TValue>);
const
N = 1000000;
var
i: Integer;
Counts: TArray<Integer>;
Range: Integer;
begin
SetLength(Counts, Length(Combinations));
for i := 1 to N do
inc(Counts[IndexOf(GetSample)]);
Range := MaxIntValue(Counts) - MinIntValue(Counts);
Writeln(Name);
Writeln(StringOfChar('-', Length(Name)));
Writeln(Format('Range = %d, N = %d', [Range, N]));
Writeln;
end;
begin
Randomize;
InitialiseCombinations;
CheckCounts('Heffernan', GetSampleHeffernan);
//CheckCounts('Van Dien', GetSampleVanDien);
CheckCounts('Lama', GetSampleLama);
Readln;
end.
The output, from one particular run, is:
Heffernan --------- Range = 620, N = 1000000 Lama ---- Range = 200192, N = 1000000
The Van Dien variant is commented out at the moment since it produces invalid values.
OK, I debugged and fixed the Van Dien variant. The test and results now look like this:
{$APPTYPE CONSOLE}
uses
System.SysUtils,
System.Math,
Generics.Collections;
type
TValue = record
a, b, c, d: Integer;
procedure Write;
class operator Equal(const lhs, rhs: TValue): Boolean;
end;
procedure TValue.Write;
begin
Writeln(a, ' ', b, ' ', c, ' ', d);
end;
class operator TValue.Equal(const lhs, rhs: TValue): Boolean;
begin
Result := (lhs.a=rhs.a) and (lhs.b=rhs.b) and (lhs.c=rhs.c) and (lhs.d=rhs.d);
end;
var
Combinations: TArray<TValue>;
procedure InitialiseCombinations;
var
a, b, c, d: Integer;
Value: TValue;
List: TList<TValue>;
begin
List := TList<TValue>.Create;
try
for a := 2 to 7 do
for b := 2 to 7 do
for c := 2 to 7 do
begin
d := 22 - a - b - c;
if InRange(d, 2, 7) then
begin
Value.a := a;
Value.b := b;
Value.c := c;
Value.d := d;
List.Add(Value);
end;
end;
Combinations := List.ToArray;
finally
List.Free;
end;
end;
function GetSampleHeffernan: TValue;
begin
Result := Combinations[Random(Length(Combinations))];
end;
function GetSampleVanDien: TValue;
const
TOTAL = 22;
VALUE_COUNT = 4;
MIN_VALUE = 2;
MAX_VALUE = 7;
var
Values: array[0..VALUE_COUNT-1] of Integer;
Shortage: Integer;
Candidates: TList<Integer>;
ValueIndex: Integer;
CandidateIndex: Integer;
begin
Assert(VALUE_COUNT * MAX_VALUE >= TOTAL, 'Total can never be reached!');
Assert(VALUE_COUNT * MIN_VALUE <= TOTAL, 'Total is always exceeded!');
Candidates := TList<Integer>.Create;
try
for ValueIndex := 0 to VALUE_COUNT-1 do
begin
Values[ValueIndex] := MIN_VALUE;
Candidates.Add(ValueIndex);
end;
Shortage := TOTAL - VALUE_COUNT * MIN_VALUE;
while Shortage > 0 do
begin
CandidateIndex := Random(Candidates.Count);
ValueIndex := Candidates[CandidateIndex];
inc(Values[ValueIndex]);
if Values[ValueIndex] = MAX_VALUE then
Candidates.Delete(CandidateIndex);
dec(Shortage);
end;
finally
Candidates.Free;
end;
Result.a := Values[0];
Result.b := Values[1];
Result.c := Values[2];
Result.d := Values[3];
end;
function GetSampleLama: TValue;
type
TRandomValues = array[1..4] of Integer;
var
IntSum: Integer;
Values: TRandomValues;
begin
// initialize a helper variable for calculating sum of the generated numbers
IntSum := 0;
// in the first step just generate a number in the range of 2 to 7 and store
// it to the first integer element
Values[1] := RandomRange(2, 7);
// and increment the sum value
IntSum := IntSum + Values[1];
// as the next step we need to generate number, but here we need also say in
// which range by the following rules to ensure we ever reach 22 (consider, if
// the 1st number was e.g. 3, then you can't generate the second number smaller
// than 5 because then even if the next two numbers would be max, you would get
// e.g. only 3 + 4 + 7 + 7 = 21, so just use this rule:
// Values[1] Values[2]
// 2 6..7
// 3 5..7
// 4 4..7
// 5 3..7
// 6..7 2..7
Values[2] := RandomRange(Max(2, 8 - Values[1]), 7);
// and increment the sum value
IntSum := IntSum + Values[2];
// if the third step we need to generate a value in the range of 15 to 20 since
// the fourth number can be still in the range of 2 to 7 which means that the sum
// after this step must be from 22-7 to 22-2 which is 15 to 20, so let's generate
// a number which will fit into this sum
Values[3] := RandomRange(Max(2, Min(7, 15 - IntSum)), Max(2, Min(7, 20 - IntSum)));
// and for the last number let's just take 22 and subtract the sum of all previous
// numbers
Values[4] := 22 - (IntSum + Values[3]);
Result.a := Values[1];
Result.b := Values[2];
Result.c := Values[3];
Result.d := Values[4];
end;
function IndexOf(const Value: TValue): Integer;
begin
for Result := 0 to high(Combinations) do
if Combinations[Result] = Value then
exit;
raise EAssertionFailed.Create('Invalid value');
end;
procedure CheckCounts(const Name: string; const GetSample: TFunc<TValue>);
const
N = 1000000;
var
i: Integer;
Counts: TArray<Integer>;
Range: Integer;
begin
SetLength(Counts, Length(Combinations));
for i := 1 to N do
inc(Counts[IndexOf(GetSample)]);
Range := MaxIntValue(Counts) - MinIntValue(Counts);
Writeln(Name);
Writeln(StringOfChar('-', Length(Name)));
Writeln(Format('Range = %d, N = %d', [Range, N]));
Writeln;
end;
begin
Randomize;
InitialiseCombinations;
CheckCounts('Heffernan', GetSampleHeffernan);
CheckCounts('Van Dien', GetSampleVanDien);
CheckCounts('Lama', GetSampleLama);
Readln;
end.
Heffernan --------- Range = 599, N = 1000000 Van Dien -------- Range = 19443, N = 1000000 Lama ---- Range = 199739, N = 1000000
And just to ram it home, here are some plots of empirical probability mass function of the various distributions:
OK, now I fixed @TLama's code. It was using RandomRange incorrectly. The documentation states:
RandomRange returns a random integer from the range that extends between AFrom and ATo (non-inclusive).
The key is that the range is defined as a closed-open interval. The value returned is in the range [AFrom..ATo), or expressed with inequality signs, AFrom <= Value < ATo.
But @TLama's code is written on the assumption that the interval is closed at both ends. So the code can readily be fixed by adding 1 to the second parameter of each call to RandomRange. When we do that the output looks like this:
Heffernan --------- Range = 587, N = 1000000 Van Dien -------- Range = 19425, N = 1000000 Lama ---- Range = 79320, N = 1000000
And the empircal PMF plot becomes:
The bottom line in all this is that sampling is difficult to get right if you care about the distribution.
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