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Kimiko and Miko are playing a game in which each girl rolls a number cube. If the sum of the numbers is a prime number, then Miko wins. Otherwise Kimiko wins. Determine the sample space. Then determine whether the game is fair.

User Jackieyang
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Final answer:

The sample space for this game consists of all possible outcomes when two players each roll a number cube. The game is not fair because one player has a higher probability of winning than the other.

Step-by-step explanation:

The sample space for this game can be determined by listing all possible outcomes when Kimiko and Miko each roll a number cube. The number cube has 6 faces numbered 1 to 6. So, the sample space is:
S = {1+1, 1+2, 1+3, 1+4, 1+5, 1+6, 2+1, 2+2, 2+3, 2+4, 2+5, 2+6, 3+1, 3+2, 3+3, 3+4, 3+5, 3+6, 4+1, 4+2, 4+3, 4+4, 4+5, 4+6, 5+1, 5+2, 5+3, 5+4, 5+5, 5+6, 6+1, 6+2, 6+3, 6+4, 6+5, 6+6}
After simplifying, the sample space is:
S = {2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12}

Next, we need to determine whether the game is fair. To do this, we can find the probability of each player winning. Miko wins if the sum of the numbers is a prime number. The prime numbers in the sample space are {2, 3, 5, 7, 11}. Since there are 5 prime numbers and 36 possible outcomes, the probability of Miko winning is 5/36.

Kimiko wins if the sum of the numbers is not a prime number. So, the probability of Kimiko winning can be calculated by subtracting Miko's probability from 1:
P(Kimiko winning) = 1 - P(Miko winning)
P(Kimiko winning) = 1 - 5/36
P(Kimiko winning) = 31/36

Since Kimiko has a higher probability of winning (31/36) compared to Miko (5/36), the game is not fair.

User Neron
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