The game of Match is played with a special deck of 27 cards. Each card has...

 

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The game of Match is played with a special deck of 27 cards. Each card has three attributes: color, shape and number. The possible color values are {red, blue, green}, the possible shape values are {square, circle, heart}, and the possible number values are {1, 2, 3). Each of the 3 * 3 * 3 = 27 possible combinations is represented by a card in the deck. A match is a set of 3 cards with the property that for every one of the three attributes, either all the cards have the same value for that attribute or they all have different values for that attribute. For example, the following three cards are a match: (3, red, square), (2, blue, square), (1, green, square). (a) If we shuffle the deck and turn over three cards, what is the probability that they form a match? Hint: given the first two cards, what is the probability that the third forms a match? (b) If we shuffle the deck and turn over n cards where n ≤ 27, what is the expected number of matches, where we count each match separately even if they overlap? Note: The cards in a match do not need to be adjacent! Is your expression correct for n = 27? The game of Match is played with a special deck of 27 cards. Each card has three attributes: color, shape and number. The possible color values are {red, blue, green}, the possible shape values are {square, circle, heart}, and the possible number values are {1, 2, 3). Each of the 3 * 3 * 3 = 27 possible combinations is represented by a card in the deck. A match is a set of 3 cards with the property that for every one of the three attributes, either all the cards have the same value for that attribute or they all have different values for that attribute. For example, the following three cards are a match: (3, red, square), (2, blue, square), (1, green, square). (a) If we shuffle the deck and turn over three cards, what is the probability that they form a match? Hint: given the first two cards, what is the probability that the third forms a match? (b) If we shuffle the deck and turn over n cards where n ≤ 27, what is the expected number of matches, where we count each match separately even if they overlap? Note: The cards in a match do not need to be adjacent! Is your expression correct for n = 27?