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 36Proc 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 [1]. 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.