Disjoint subsets

Probability that two k-sized independent subsets A,B from set of size n are disjoint

Set size n:     #trials of each k=1,⌊n/2⌋:

k (A∩B=∅) n-kCk/nCk=

Try:
n=10 #trials=1000

n=100 #trials=1000 

Probability that two k-sized subsets A,B are disjoint is n-kCk / nCk

n=10
k n-kCk   / nCk
1 9C1= 9 /  10 = .9
2 8C2=28 /  45 = .622
3 7C3=35 / 120 = .292
4 6C4=15 / 210 = .071
5 5C5= 1 / 252 = .004


n=100
k n-kCk   / nCk
1 99C1= 99 /  100 = .99  99% chance they are disjoint (not the same element)
2 98C2= 4753/4950= 0.9602
3 97C3= 147440/161700= 0.9118
4 96C4= 3321960/3921225= 0.8472
5 95C5=  = 0.7696
6 94C6=  = 0.6830
7 93C7=  = 0.5918
8 92C8=  = 0.5002  50-50 whether they are disjoint or not
9 91C9=  = 0.4120
10 90C10=  = 0.3305
11 89C=  = 0.2579
12 88C12=  = 0.1955
13 87C13=  = 0.1439
14 86C14=  = 0.1027
15 85C15=  = 0.0710
...
20 80C20=  = 0.0066  99% chance they have common element(s)
...

An element of an n-set is in half (2n-1) of all (2n) subsets.
So the probability that it is in a randomly chosen subset is ½.

So the probability that it is in two randomly chosen subsets is 1/4.
and the probability that it is in neither of the two subsets is 3/4.
Same for each of the n elements, thus the probability that two subsets
are disjoint is (3/4)n.