[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, there is 1 way to throw 2, 2 ways to throw 3 (2,1 & 1,2) 3 ways to throw 4 (1,3; 3,1; 2,2) and so on.  The following code provides some 
routines for emulating 1 or 2 dice using the simple rand() function. Paste this code into wish as it uses a Tk based histogram to plot the frequency.

 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
	}
	histogram [array get bins] 300 250
	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.
Proc histogram produces a Tk plot of the frequency of landing on each square.
You will find that the number of times your throw lands on a particular square increases, then decreases, and then oscillates a little. It is remarkable that the rise and fall for squares up to 13 are almost linear. This is due to the probability of each throw increasing and decreasing linearly rather than in a nice bell shaped curve you might have expected [http://www.mathreference.com/pr,distf.html]. However landing at 7 can be done by throwing a 7 on first throw, OR by throwing 5&2, 2&5, 4&3, 3&4 or 2&2&3 or 2&3&2 or 3&2&2 (taking longer to get there but increasing the number of ways of reaching 7); this increases the probability of landing on 7 above the simple linear probability.

 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

... this output is numerical results used in the histogram.


[Category Mathematics]