More Memory Game

This post is an attempt to answer the question I posed in a prior post about the Memory Game. If you are not familiar with the Memory Game, you can read that post to get familiar with the rules and terminology. The question:

With the second part of your turn, if you do not have a 100% chance of getting a pair is it better to flip an already known tile or try to flip an unknown tile?

As I explained in my prior post on Memory, there are two ways to score. The first method of scoring is to flip over your first tile and already know the position of its match on the board. The second method of scoring is to flip over your first tile, not know the position of its match, and then randomly guess and find its match. My question asks should you forgo the second method of scoring pairs? 

Last post I mentioned how I created a table that gives the % chance of a player scoring a pair based on how many tiles are left in the game (n) and how many tiles are known at the beginning of the turn (m). By analyzing that table it quickly became clear that it is much more likely to score a pair by the first method than by the second method. If m=0 then you are more likely to score by the second method--as it is impossible to score by the first method--and if m=1 you are equally likely to score a pair by either method. For all other possible values of m and n, you are much more likely to score by using the first method; 68% - 98% of pairs happen by means of the first method. As m increases the higher the likelihood is that you will score using the first method.


Intuitively it makes sense to flip one tile--and if no match is known--pass your turn (not try to score using the second method) in some circumstances. Suppose there are 10 tiles on the board. 3 have already been flipped and are known. Player 1 (P1) flips an unknown tile. If that tile does not match with one of the known tiles, he can then flip one of the six unknown tiles. Let's consider his possible outcomes:

• 1/6 Chance: Flips a match to his existing tile and scores a pair (Favorable Result)
• 2/6 Chance: Flips an unknown card that does not match one of the known tiles (Neutral Result)
• 3/6 Chance: Flips a match to one of the known tiles (Unfavorable Result)

P1 has a 1/6 chance of getting that pair. However P1 has a 3/6 chance of flipping a match to one of the known tiles. If he flips one of those tiles, Player 2 (P2) will proceed to score a pair with 100% certainty. 


P1 flips the first tile. If no match is known at that time he has two choices: try to score using the second method or pass his turn (flip an already known tile which will not result in a match). If he uses the second method, he will either flip a Favorable Result, Neutral Result, or Unfavorable Result. The chance of a Favorable Result is less likely than the chance of an Unfavorable Result if m is greater than 1. The chance of a Favorable Result = 1/(n-m-1) while the chance of an Unfavorable Result = m/(n-m-1). Basically by attempting to score using the 2nd method, you are more likely to give your opponent a pair than to score a pair yourself.


A few other considerations. The Neutral Result --while maybe not immediately harmful to P1--does increase m by 1 for P2, thus giving him a higher chance of scoring a pair. This is partially offset by P1 possibly also having a higher m on his turn but in general raising m is more helpful for your opponent than for you. If you begin the turn in a situation where m = n/2 - 1 it may be helpful to try to score using the second method. If m = n/2-1 at the beginning of the turn and you fail to score using the first method, your opponent will begin his turn with a 100% chance of scoring anyway. However by trying the second method, even if you fail to score you will have a 100% chance of scoring on your next turn as you will begin your next turn with m = n/2.