S1 Question: Difficult probability

This is from a Delphis paper- I am quite confused at question b). Surely if the card is TRANSFERRED from Group A to group B then the probabilities would be:

(5/29) * (7/32)?

Yet the mark scheme says:

(5/30 * (8/32), implying that the number of cards in pack B has increased, but not decreased in A. Also implies that the card from the question has first digit 1, when it is actually first digit 2?

I'm probability just being really dense but does any one have any idea what to do here?
Thanks, much appreciated.

Can you post the table of information that goes with this question?

This is from a Delphis paper- I am quite confused at question b). Surely if the card is TRANSFERRED from Group A to group B then the probabilities would be:

(5/29) * (7/32)?

Yet the mark scheme says:

(5/30 * (8/32), implying that the number of cards in pack B has increased, but not decreased in A. Also implies that the card from the question has first digit 1, when it is actually first digit 2?

I'm probability just being really dense but does any one have any idea what to do here?
Thanks, much appreciated.

Event $A_{1}$ refers to the original situation when you take a card from pack A and look at its first digit - so you're looking at the original number of cards in A. If you now put this card in B and look at another card in A, then A is now smaller!

Yes but the value for A isn't smaller! It's the same?

I think you're confusing yourself about what's happening here (or confusing me )

We take a card from A. Its first digit might be 1, with probability 5/30; or it might be 2, with probability 7/30; or it might be one of the other possibilities.

When they say "write down $P(A_2)$ ", they're just asking you to write down the relevant probability - they're NOT telling you that the first digit is 2.

Now the card is put in pack B (so B is larger) and we draw a card from B. Its first digit could be 1, with probability $B_1$, or one of the other probabilities.

So when they ask you about $P(A_1 \cap B_1)$ they're asking you to combine the probability from the 1st draw from A with the probability from the 1st draw from B. You're not taking another card from the (smaller) pack A!

Does this help?

Thanks for the help! I get it but it's still annoyingly confusing- I am just going to pray that anything remotely like this doesn't come up in the exam tomorrow!