What is the probability of #4

[w0lfshad3]

What is the probability of the player matching exactly 3 numbers from a 10 number lottery (0...9) w/o repetition, given the conditions:
- player and lottery pick 6 numbers;
- player always has a repeated number;
- lottery always has a repeated number.
for example:

123455
123455

Here there's many matches of 3. Because it's a match of 6/6 it means that there are C(6,3)=20 matches of 3... I think

Note. Legit choice:
123455
112345
It also means the player matched 4 numbers while picking only 3, which means C(4,3) ways to match a 3 number... match

 

[w0lfshad3]

Actually the point is only the number matchings on the lottery using just 3 player numbers.

This is a conditional probability problem.

I experimented on matching all numbers between ABCD;EE and ABCD;EE
Hypothetic (calculated) probability: 1/1260

Real probability: ~1.75/1260
0.0013608333 experiment
0.00138888(8) guess = ~1.75/1260=1/720=1/6!=1/P(10,3)?=1/(1x2x3x4x5x6)

Anyone knows how to calculate this properly?