Calculating amount of information gained by getting partial data

Question

Set up: Let's say I have a bag with 12 notes containing strings of text made up of the letters A-L.

Action: Someone pulls one note out from that hat, then at random they choose 2 letters written on their note, and tell those letters to me.

Goal: I'm trying to calculate the average amount of information I get by knowing any 2 letters off their note.

Case study:

To make it super flushed out, let's assume these are the notes in the hat:

1. ACEFHJL
2. ABDFGIK
3. BCEGHJL
4. ACDFHIK
5. BDEGIJL
6. ACEFHJK
7. BDFGIKL
8. ACEGHJL
9. ABDFHIK
10. BCEGIJL
11. ACDFHJK
12. BDEGIKL

Example 1:

If for instance, this person tells me their note contains the letters A and B, I know they are either holding note number 2, or note number 9, since those are the two notes containing both A and B. This therefore, cuts the space of possibilities from 12 down to 2, equivalent to about 2.58 bits of information.

Example 2:

If on the other hand, they tell me their note contains the letters A and F, it would mean that the note they are holding is number 1, 2, 4, 6, 9, or 11, since they all contain both A and F. This in turn only cuts the space in possibilities in half, thus providing only 1 bit of information.

Recap:

So, we see that knowing some 2-letter combinations (like AB) provide more information than others (like AF). I would like calculate the average amount of information I get by knowing any 2 letters? That is to say, if we repeated this exercise a million times, each time, this person would shuffle the notes in the hat and pull out one. Then at random tell me 2 letters off of that note. How much information will I gain on average?

Flushing it all out:

Here's another example. Let's say this time I have a bag with the following notes in it:

1. ACEG
2. ABDF
3. BCEG
4. ACDF
5. BDEG
6. ACEF
7. BDFG

Below is a table with the information gained by knowing the selected note contains any 2 letters

Letter combo	p(E)	I(E)	Appears in:
AB	0.14 (1/7)	2.84	2
AC	0.43 (3/7)	1.22	1, 4, 6
AD	0.29 (2/7)	1.79	2, 4
AE	0.29 (2/7)	1.79	1, 6
AF	0.43 (3/7)	1.22	2, 4, 6
AG	0.14 (1/7)	2.84	1
BC	0.14 (1/7)	2.84	3
BD	0.43 (3/7)	1.22	2, 5, 7
BE	0.29 (2/7)	1.79	3, 5
BF	0.29 (2/7)	1.79	2, 7
BG	0.43 (3/7)	1.22	3, 5, 7
CD	0.14 (1/7)	2.84	4,
CE	0.43 (3/7)	1.22	1, 3, 6
CF	0.29 (2/7)	1.79	4, 6
CG	0.29 (2/7)	1.79	1, 3
DE	0.14 (1/7)	2.84	5
DF	0.43 (3/7)	1.22	2, 4, 7
DG	0.29 (2/7)	1.79	5, 7
EF	0.14 (1/7)	2.84	6
EG	0.43 (3/7)	1.22	1, 3, 5
FG	0.14 (1/7)	2.84	7

How do I compute the average information gained by selecting a random row from this table?