Adding probabilities: There are 10 guys and 10 girls. Each guy is a perfect match wit

I used to be good in math but school was long time ago lol
So, I'm watching this dating show and trying to calculate probabilities.
There are 10 guys and 10 girls. Each guy is a perfect match with only one girl. We don't know those matches yet.
All 10 couple up and we get the first clue: 4 couples are a match, the other 6 are not. We don't know who the 4 correct matches are. So probability for each couple is 0.4
All 10 couple up and we get the second clue: 2 couples are a match, the other 8 are not. Probability of 0.2

Now, 3 couples were the same in the first and the second clue. Those have a higher probability to be a match, right? But 0.4 + 0.2 = 0.6 seems not to be right...
I got stuck!
Help!

leeloo said:

There are 10 guys and 10 girls. Each guy is a perfect match with only one girl. We don't know those matches yet. All 10 couple up and we get the first clue: 4 couples are a match, the other 6 are not. We don't know who the 4 correct matches are. So probability for each couple is 0.4
All 10 couple up and we get the second clue: 2 couples are a match, the other 8 are not. Probability of 0.2
Now, 3 couples were the same in the first and the second clue. Those have a higher probability to be a match, right? But 0.4 + 0.2 = 0.6 seems not to be right...

Click to expand...

In thirty years of teaching probability theory, I have never read a more confused statement.

Here is my take. There is one perfect matching out of \(\displaystyle 10!=3628800\) possible pairings.
You must understand derangements. That is a good reference. But I absolutely hate the notation \(\displaystyle !n\).
I use the notation \(\displaystyle \mathscr{D}(n)\approx \frac{n!}{e},~n>3\).
So there are \(\displaystyle \binom{10}{4}\mathscr{D}(6)\) ways to pair this group with exactly four perfect pairing.
Here is a webpage to help with calculations of derangements.

But I have no idea what you are trying to do.

pka, thank you so much for your response! I looked into derangements and I don't see a way to solve it that way.
I also read the rules for posting (apologies: should have done that beforehand) and I see, I'm supposed to post the whole problem instead of extracting only the part I'm struggling with. The attachment shows what I have so far:



The way I see it, there are 100 possible couples. 10 x 10 (since a guy can't pair up with another guy and the same for girls). That's what the matrix is for. As a start, I put a probability of 1 out of 10 (0.1) for each couple.

1. Week: We find out that Chris T and Shanley are not a match. I mark that in the matrix which leaves all other pairings with Shanley a probability of 1 out of 9 instead of 10 (0.11). The same applies for Chris T.

The blue box on the top right side shows the couples from the first week. 2 out of 10 are matches. (In week 2 we find out that John and Simone are not a match, so I adjusted to 2 out of 9 are a match = 0.22). Putting this information into the matrix.

2. Week: We find out that John and Simone are not a match. I mark that in the matrix which leaves all other pairings with Simone a probability of 1 out of 9 and the same for John.

The second blue box on the right shows the couples from the second week. 4 out of 10 are matches. A probability of 0.4. Putting this information into the matrix.

3. Week: We find out that Ethan and Jessica are not a match. Adding this information into the Matrix and adjusting all other pairings with Ethan or Jessica.

The third blue box on the right shows the couples from the third week. 2 out of 10 are matches. Putting this information into the matrix.

This is where I get stuck: Ethan and Amber were a couple in the second week (0.4 probability) and in the third week (0.2 probability). Logically, I would expect their sum probability to be higher than 0.4 but I don't know how to exactly calculate it

Adam and Britney were a couple in the first week (0.22) and in the third week (0.2).

Jason Dre and Ashley were a couple in the second (0.4) and third (0.2) week.

Ryan and Kayla were a couple in the second (0.4) and third (0.2) week.

Dillon and Calicia were a couple in the first (0.22) and third (0.2) week.

(Or am I maybe mixing too many variables?)

It reminds me of the game Mastermind where you have to find out the right color pin and its right placement.
The difference here is that you don't need to find the right placement, just the right color (=couple?)