Two players-Player A, who goes first; and Player B, who goes second-are playing

Question

Two players-Player A, who goes first; and Player B, who goes second-are playing a game of Memory. In Memory, all cards have a specific color, but every card begins "face down," so that only the back of the card is seen (all backs look the same). On a player's turn, she selects one card and turns it over; then she selects another card and turns this one over. If the colors match, she keeps those cards (called a set) and gets another turn. If the colors do not match, she turns them back over (so they are face down again) and the other pl that both players have perfect memories, meaning that they always remember which cards were previously turned up and where they are. Also assume that both players are wise, choosing a match if possible. (So assume that when a player reveals her first card, if its match has already been turned over on a previous turn, she will choose it and get a set. Also as- sume that if two matching cards are known but face down, the next player will choose them and get a set. Also assume that if the known face-down cards do not contain a match, the next player will choose a different card first.) Remember: Get a match -go again! ayer goes. Assume This game of Memory is a bit different. There are only 5 cards: 2 cards are blue, and 2 cards are green, and 1 card is red. Thus the red card has no match. If you are playing this game, is it better to go first or second? Answer by finding the following probabilities: the first player (Player A) gets two sets and wins; the second player (Player B) gets two sets and wins; each player gets one set, which results in a tie.

Explanation / Answer

Given 5 cards, blue - 2, green - 2, red 1

Probability of player A gets two sets and wins = 2/5 * 1/4 + 2/3 * 1/2 = 13/30

Probablity of player B gets two sets and wins = 1/5 * 3/4 + 3/5 (since if the first set does not match the next player chooses from different set of cards) * 1/4 + 2/3*1/3 = 19/30

Probablity of each player gets one set and wins = 1/5 * 3/4 + 3/5 * 1/4 + 2/3 * 1/3 + 2/3* 1/3 = 29/30

.

As the probablity of player b winning the match (19/30) > the probablity of player a wining the match (13/30)

therefore it is best to go for second

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