Batch scripting Issues with random, probability or a bug

Question

Please excuse any mistakes, this is my first post! Recently I was bored and was trying to make a batch script clone of an arcade game I once saw: http://antikrish.com/2013/01/28/whittakers-ascot-gold-cup-horse-racing-arcade-game/

Rules

I couldn’t remember the rules and wanted it to be fair so made my own.

• Each horse has a number between 1 and 6
• This number (x) is the chance of the horse moving each turn (1/x)
  • Horse 1 = 1/1 so moves every turn
  • Horse 3 = 1/3 so moves every three turns on average
  • Horse 6 = 1/6 so moves every six turns on average
• This number (x) also dictates how are along the board the horse moves
  • Horse 1 = 1 space per move
  • Horse 3 = 3 spaces per move
  • Horse 6 = 6 spaces per move
• The game board is 60 spaces long, the first horse/s to reach 60 are the winners (draws are allowed). 60 was chosen as a goal as it is a common multiple of 1, 2, 3, 4, 5 and 6.
• The idea of this is that it should take each horse approximately 60 turns to reach the end and win.
  • Horse 1 => 60 spaces / 1 moves = 60 turns, 60 turns / (1/1) = 60 turns
  • Horse 3 => 60 spaces / 3 moves = 20 turns, 20 turns / (1/3) = 60 turns
  • Horse 6 => 60 spaces / 6 moves = 10 turns, 10 turns / (1/6) = 60 turns

Problem

I ran the game and everything seemed good! Except after a while I noticed Horse 1 almost never seemed to win. Now this could be explained with probability but I wanted to check, so I altered my code to keep track of the results over time and saw that the higher numbered horses were winning consistently more often!

I’ve uploaded a graph (https://i.sstatic.net/gSzqV.png) showing the results of the winners of three sets of 1000 races, where draws are allowed (and count as a win for all that finished). Now this behaviour isn’t necessarily a bad thing and means odds can be attached to each horse, however I am confused as to why it’s happening.

Reasons

In my mind there are three reasons for why this behaviour may be occurring:

• I have a bug in my program. I’ve included the code below if you're interested.
• Batch random is attached to time, and this is screwing the results making the higher numbers more likely. To test this I collected three sets of 1000 random numbers between 1 and 6 and the results seem alright.
  • Note I varied the time between the numbers being generated:
  • REM – no time gap between generation
  • 0 Sec – 0 seconds on timeout (so a brief pause)
  • 1 Sec – 1 second on timeout
• This is an attribute of probability and payoffs, and the logic I’ve used is flawed for a fair game.

So, can anyone enlighten me as to why this may be happening?

Thanks in advance!

Here is the code if helpful:

@ECHO OFF
COLOR 0B
setlocal EnableDelayedExpansion
set total = 0
set totalscore[0]=0
set totalscore[1]=0
set totalscore[2]=0
set totalscore[3]=0
set totalscore[4]=0
set totalscore[5]=0
set totalscore[6]=0

:START
set /a total = total + 1
set score[0]=0
set score[1]=0
set score[2]=0
set score[3]=0
set score[4]=0
set score[5]=0
set score[6]=0
set winner=0

:LOOP
cls
echo Race Number: %total%
for /l %%n in (1,1,6) do (
    set /a num=!random! %%%%n + 1
    IF !num! == %%n set /a score[%%n]=!score[%%n]! + !num!
    <NUL set /p=%%n: 
    for /l %%x in (1, 1, !score[%%n]!) do <NUL set /p=^|
    set /a sum=60 - !score[%%n]!
    for /l %%x in (1, 1, !sum!) do <NUL set /p=_
    echo F
) 
for /l %%n in (1,1,6) do (
    if !score[%%n]! == 60 (
        echo winner: %%n
        set /a totalscore[%%n]=!totalscore[%%n]! + 1
        set winner=1
    )
) 

echo.
echo Previous Results
for /l %%n in (1,1,6) do (
    echo %%n: !totalscore[%%n]!
) 

if %winner%==1 GOTO END
timeout /t 0 /nobreak > NUL

GOTO LOOP

:END
echo.
pause

GOTO START

EDIT I think its a probability quirk, as Horse 6 has a chance to finish after 10 turns, where all other horses physically can't. This adds up for each horse, reducing the horses below it's chances (i.e horse 1 will only win if the other's all under perform).

Thanks for the responses :)

Answer 1

The issue comes down to probability!

If we simplfy the game down to 6 moves:

• Horse 1 will finish in 6 moves, however horse 6 has 5 attempts to beat it at 1/6 chance each try.

• The odds of horse 6 NOT finishing = (5/6)^5 = 0.40 Therefore the odds of horse 6 finishing in 5 moves or less is 0.6

If we do this for each of the horses, poor horse 1 never stood a chance!