The Double Slit Experiment

BASH, 396 - 11 = 385 bytes

E='echo -en';$E "\e[2J\e[99A";while :;do sleep 0.01;for i in `seq $((($1+1)/2)) -1 1`;do p=$(((($1+1)/2 - $i)));[ $p -lt 0 ]&&p=$((-$p));p=$((2**(p+1)));if [ $RANDOM -lt $((32768/$p)) ];then [ $(($RANDOM%2)) -eq 1 ]&&i=$((($1+1)-i));sector=$(((i*2-1)-1));C=`tput cols`;R=`tput lines`;SS=$((C/($1*2-1)));SX=$((SS*sector));X=$((SX+(RANDOM%SS)));Y=$((RANDOM%R));$E "\e[$Y;${X}H*";break;fi;done;done

Unfortunately I cannot demonstrate this on TryItOnline because of the endless loop & ANSI escape sequences that move cursor, but you can still copy-paste it into your terminal!

Unminified version:

E='echo -en'
$E "\e[2J\e[99A"

while :
do
    sleep 0.01
    for i in `seq $((($1+1)/2)) -1 1`
    do
        p=$(((($1+1)/2 - $i)))
        [ $p -lt 0 ] && p=$((-$p));
        p=$((2**(p+1)))
        if [ $RANDOM -lt $((32768/$p)) ]
        then
            [ $(($RANDOM%2)) -eq 1 ] && i=$((($1+1)-i));
            sector=$(((i*2-1)-1))
            C=`tput cols`
            R=`tput lines`
            SS=$((C/($1*2-1)))
            SX=$((SS*sector))
            X=$((SX+(RANDOM%SS)))
            Y=$((RANDOM%R))
            $E "\e[$Y;${X}H*"
            break
        fi
    done
done

Python 2, 207 200 bytes

There is a method to this madness, I promise. Follows the probability interpretation I commented in the OP.

Edit: -7 bytes through some clever lazy evaluation (and removing some signs)

import time  # not counted for byte total
import random as R,curses as C
r=R.randint
c=C.initscr()
h,w=c.getmaxyx()
n=input()
w/=2*n-1
while 1:
 all(r(0,1)or c.addch(r(0,h-1),(i*(2-4*r(0,1))+n)*w-r(1,w),42)for i in range(n/2+1))
 c.refresh()
 time.sleep(0.1)  # not counted for byte total

Mathematica, 231 bytes

(R=RandomInteger;p=20(#+1)+10;s=Array[0&,{20,6p-3}];i=(#+1)/2;Monitor[While[1<2,y=RandomChoice[Join[q=Riffle[Array[2^#&,i,0],Table[0,i-1]],Reverse@Most@q]->Array[Range[4#+1]&,i,0][[i]]];s[[R@19+1,10y-R@9]]=1;s],Grid[s//. 0->" "]])&

input

[5]

output