Average number of days to see all possible cards

Solution 1:

If you draw with replacement instead, then this is the coupon collector's problem, which has a simple solution and is still a good approximation of your situation (since drawing the same card twice on a given day is unlikely).

The expected number of draws to get all coupons from a set of $n$ is $n\sum_{k=1}^n\frac1k\approx n \log n + \gamma n + \frac12$, where $\gamma$ is Euler's constant. For $n=600$, this is about $4185$ draws, or $2092.5$ days at two draws/day.

We can adjust this to simulate drawing without replacement on each day. Drawing two cards without replacement is equivalent to drawing with replacement until we've seen two different cards. If we do this, each draw after the first one has an $\frac{n-1}n$ chance of being different from the first, so the expected number of draws per day is $1+\frac n{n-1}=2+\frac1{n-1}$. At this rate, it takes about $4185/(2+\frac1{599})\approx2090.75$ days to see all the cards.

Solution 2:

Let $E_k$ be the expected number of days needed to see all cards when $k$ cards are unseen, and $N$ be the total card count. Then $E_{-1}=E_0=0$ and $$E_k=1+\left(\binom{N-k}2E_k+k(N-k)E_{k-1}+\binom k2E_{k-2}\right)\Big/\binom N2$$ $E_N$ may be easily computed using this recurrence. For $N=600$ this works out to

8811680562792459942398946258747101711876096393133977154769476674327738047674364897328983432691011862688405751916731357166104734388331528355089282222739184014648126943968991084493699738362309278580044419512610098361119484129038113839696652596911039150120983887537144163875349874748255808223097625902743818172423335386062006583188763439060477540352487897399170791765802797781451481218366968559229380559865123297655357640689131813797606942310981351167391816229887003702468658214065550812178233069632038668626978661/4214102157325039618584833864326203795997776444177870231182813656392729795772789605802732077307165602541014302980322730432077482215058464516793042569634244594830162890533802603313182085689051070210521045596971279589537297216582507288609368821092057740008461735909598505487560158947404502130949413085628364157149213652932303703915433272481718831240417233576842751450658118984157039823591959867158637197417212761707815524914107310227771047305213850150648656068313470561815204614063875022693603836462568954334528

which is equal to $2090.998327\dots$