Memory

There's an app on my phone that is a game of memory with 16 cards.  Each cars has a point value.  When you pair, the points go in your bank which can then be redeemed for free items.  Pretty sweet deal!
You can miss four pairs.  On the fifth miss, the game is over.  Being a super nerd, I'm working on figuring out both the probability of clearing the board (matching all 8 pairs before having 5 misses) and the expected value of a play.
The 5 plays accumulate over time.  Last night at dinner I introduced the game to a colleague, who started with 4 plays.  I advised him to wait for the fifth play but grew played any way.  He lost. No points.    It got me thinking of the big difference between 4 and 5 plays, as well as the difference between 5 and 6 plays.  At 4 plays, you are guaranteed nothing, at 5 plays you are guaranteed only a single pair and at 6 you are guaranteed to clear the board.  Someone put some serious math into these probabilities!  Well done app maker!