Average time taken to collect at least 1 category of cards from 10 cards with replacement

Question

let say there's 10 card in a box, which categorize to:

Spade - A, 2, 3, 4, 5

Heart - A, 2, 3, 4, 5

One card is draw at a time with replacement,

What is the average time taken to collect at least one category of cards (can be either spade or heart)?

I found that this question is actually quite similar to coupon collector problem, but I how do I apply it to question above?

Answer 1

Let $E_{s,h}$ be the expected number of draws required if you have already drawn $s$ distinct spades and $h$ distinct hearts. We want to compute $E_{0,0}$.

We can do this by working backwards from the cases where we have already drawn lots of distinct cards. For example, $$E_{4,4}=1+.8E_{4,4}\implies E_{4,4}=5$$ because we have to draw a card, and $80\%$ of the time it will be a card we already have, and the other $20\%$ of the time, the game will be over.

Also, $$E_{3,4}=1+.7E_{3,4}+.2E_{4,4}$$ by similar reasoning.

Can you finish it?