[GWM]

After reading [throw a dice], I decided to look into throwing 2 dice since many games such as [monopoly] throw 2 dice on each turn.

Using 2 dice makes life more interesting.  The most likely throw is 7 - you can never throw 1 and so on.  The following code provides some 
routines for emulating 1 or 2 dice using the simple rand() function.

 proc throwdice {} {
	set ok 0
	while {!$ok} {
		set sc [expr 1+int(6*rand())]
		if {$sc<7} {set ok 1}
	}
	return $sc
 }
 proc throw2dice {} {
	lappend all  [throwdice]
	lappend all  [throwdice]
	return $all
 }
 proc throw2dicesum {} {
	set thro [ throw2dice]
	return [expr [lindex $thro 0]+[lindex $thro 1]]
 }

 #throwdice emulates a single dice of 6 sides.
 #throw2dice returns a list of the results of 2 throwdice;
 #throw2dicesum returns the sum of two dice.

 proc getfrequency {n} {
  # check how often we arrive at each 'square' from 0 with 2 dice thrown to reach square n at least.
	set i 0
	array set bins {}
	set maxbin $n
	while {$i<=$maxbin} { set bins($i) 0; incr i}	
	set i 0
	while {$i<40000} { ;# do this number of repeats to get statistical sample.
		set j 0
		set pos 0
		while {$pos<$n} { ;# continue until we reach N
			set thro [throw2dicesum]
			incr pos $thro
			if {$pos<=$n} { ;# if > n then dont count any more.
				incr bins($pos)
			}
		}
		incr i
	}
	puts "Histogram of frequency of reaching square N from origin."
	set i 0
	while {$i<$maxbin} {
		puts "position\t$i\treached\t$bins($i)\ttimes"
		incr i
	}
	return 0
 }
 global ready
 set ready 0
 proc histogram {slist wid ht} { ;# render a histogram
	global ready
	array set scores $slist
	set nms [lsort -integer [array names scores]]
	catch {destroy .h} {}
	catch {destroy .b} {}
	canvas .h -width $wid -height $ht -bg beige
  	pack .h
	set nbars [array size scores] ;#how many histogram bars
	set nax [expr $nbars/10] ;# axis spacing
	set hwid [expr $wid/ $nbars]
	set i 0
	set hmax -9999999 ;# largest y value 
	set hmin  9999999 ;# smallest y value
	while {$i<$nbars} {
		set f $scores([lindex $nms $i])
		set hmax [ expr {$f>$hmax} ? $f : $hmax]
		set hmin [ expr {$f<$hmin} ? $f : $hmin]
		incr i
	}

	if {$hmax>$hmin} {
	set i 0
	set nay 100
	while {$nay<$hmax} {
		set yp [expr $ht-0.75*$nay-20]
		.h create line  0 $yp $wid $yp -fill red
		incr nay 100
	}
	set nax 10
	while {$i<$nbars} {
		set x [expr $hwid*$i]
		set rhs [expr $x+$hwid/2]
		set f [expr $ht-20-.75*$ht*$scores([lindex $nms $i])/$hmax]
		.h create rectangle  $x [expr $ht-20] $rhs $f -fill gray
		if {[expr $i>=$nax]} {
		.h create line  $x [expr $ht-20] $x 0 -fill red
			incr nax 10
		}
		incr i
		incr x $hwid
	}
	update

	button .b  -text "Next" -command "incr ready" ;# when ready is changed, proceeds with commands
	pack .b
	}
 }
 
 getfrequency 36

Proc getfrequency finds how often you land on each square 0 to n (eg 0-36) no matter how many throws of the dice this takes.

You will find that the number of times your throw lands on a particular square increases, then decreases, and then oscillates a little.

 Histogram of frequency of reaching square N from origin.
 position	0	reached	0	times
 position	1	reached	0	times
 position	2	reached	1128	times
 position	3	reached	2177	times
 position	4	reached	3385	times
 position	5	reached	4572	times
 position	6	reached	5750	times
 position	7	reached	7411	times
 position	8	reached	6758	times
 position	9	reached	6209	times
 position	10	reached	5826	times
 position	11	reached	5689	times
 position	12	reached	5361	times
 position	13	reached	4964	times
 position	14	reached	5557	times
 position	15	reached	5835	times
 position	16	reached	5880	times
 position	17	reached	5930	times
 position	18	reached	5803	times
 position	19	reached	5784	times
 position	20	reached	5628	times
 position	21	reached	5572	times
 position	22	reached	5710	times
 position	23	reached	5723	times
 position	24	reached	5639	times
 position	25	reached	5782	times
 position	26	reached	5851	times
 position	27	reached	5776	times
 position	28	reached	5593	times
 position	29	reached	5681	times
 position	30	reached	5651	times

I used excel to display the graph, but could use the graph plotter.

[Category Mathematics]