A person is playing the card matching game concentration. There are 40 cards, 20 pairs total. All the cards are shuffled and placed at random face down. A turn consists of two moves and a move is simply turning a card face up to see what it is. If both cards are matched during a turn they are removed form the game and the turn total goes up by one. If the cards are not matched they are returned to the face down position and again, the turn total goes up by one.
My questions:
1. What is the probability of getting a perfect score (20 turns) in a game of concentration?
2. How do you go about finding this probability?
Best Answer
Let's look at a prefect game. In each turn, we will flip one of the cards and after we've seen it, we will flip the other one, so the second (fourth, sixth...) choice of cards depend on the choice of the first (third, fifth...) card.
Since there is 1 in 39 cards to complete the pair once we've selected the first card, there is a chance of $\frac{1}{39}$ of pairing the first two cards.
Once that is done, we are left with 38 cards, so if we flip one up, there is a chance of $\frac{1}{37}$ to complete the secong pair.
...
Once we arrive to the last pair of cards, if we flip one of them up, there will be just one remaining card so the chance of completeing the last pair is $\frac{1}{1}$.
So the chance of a perfect game is:
$$\prod_{k=1}^{20}\frac{1}{2k-1} = \frac{1}{319830986772877770815625} \simeq 3.12ยท10^{-24}$$