A randomly shuffled deck of cards consists of 15 cards total of 9 suits.
We'll designate the suits as A1, A2, A3, B1, B2, B3, C1, C2, & C3.
The deck contains one card each from suits A1, A2, B1, B2, C1, & C2.
It also contain 3 cards each from suits A3, B3, & C3.
The cards are dealt to 3 hands of 5 cards, each hand being dealt in sequence.
What is the average chance of any of the 3 hands containing 1 of the following 2 outcomes:
Three or more cards from any combination of A1, A2, and/or A3 suits.
All 3 cards of A3 suit.
My probability knowledge has some serious holes so please explain your answer in simple terms. Formulae are unlikely to help as I'll have trouble understanding it. Worked examples would be ideal.
Note: If you frequent Reddits probability section you will have seen this posted in a slightly different posing already. I'm cross posting to a few places because the question is driving me up the wall, it's holding a personal project up.