I have $n$ items and I pick $3$ items from it and perform some operation. I know that the success rate of this operation is $x\%$ when I have only $3$ items. But, in order to increase the success rate of the operation, I increase my $n$ value and perform the operation on three randomly chosen items under the following condition. I am able to prove this empirically. But I have been trying a long time to prove it mathematically. Can someone help with it?
Conditions: the randomly chosen tuples from the list $n$ cannot have repetitions of pairs and triples. (I am sorry if this doesn't make sense) for example:
If $n=7$, I have $I_1,I_2,I_3,I_4,I_5,I_6,I_7$ items. My chosen tuples set is
$$(I_1,I_2,I_3); \quad (I_1,I_4,I_5); \quad (I_1,I_6,I_7); \quad (I_2,I_4,I_6); \quad (I_2,I_5,I_7); \quad (I_3,I_4,I_7); \quad (I_3,I_5,I_6)$$
This graph shows the increase in success rate with increase in $n$
For example, following are the set of tuples formed when $n=7$ and $n=9$. I apply the operation on each of the tuple and declare success if any of the tuple return a true value. For the following list, we can see that the rate is increased just because of the inclusion of [1,8,9] tuple when $n=9$. I want to prove this mathematically.
-bash-4.4pythonenumerateset.py 7 [[1,2,3],[1,4,5],[1,6,7],[2,4,6],[2,5,7],[3,4,7],[3,5,6]]
−bash−4.4 python enumerate_set.py 9 [[1, 2, 3], [1, 4, 5], [1, 6, 7], [1, 8, 9], [2, 4, 6], [2, 5, 7], [3, 4, 7], [3, 5, 6]]