Corey and Yin are trying to decide who will get to choose where they go for lunch. They decide to flip a coin until it lands showing heads. If the first flip shows heads, then Corey gets to choose. If heads shows on the second flip or later, then Yin gets to choose. Is this a fair way to decide who chooses where they go lunch? Why or why not? Choose 1 answer: (A) No, Corey is most likely to get to choose. (B) No, Yin is most likely to get to choose. (C) Yes, they both have an equal probability of choosing.

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Coin Probability Analysis: Let's analyze the probability of each outcome. The coin has two sides, heads and tails, and each side has an equal chance of landing face up on any given flip. This means the probability of getting heads (H) or tails (T) on any flip is 1/2.Corey's Winning Probability: Corey wins if the first flip is heads. The probability of this happening is 1/2, since there is only one flip involved and the coin is fair.Yin's Winning Probability: Yin wins if the first flip is tails and the second flip is heads, or if the first two flips are tails and the third flip is heads, and so on. This means Yin's probability of winning is the sum of the probabilities of all these events occurring.Yin's Winning Probability Calculation: The probability that Yin wins on the second flip is the probability that the first flip is tails (1/2) AND the second flip is heads (1/2). We multiply these probabilities because we want both events to happen, which gives us (1/2) * (1/2) = 1/4.Pattern in Yin's Winning Probability: The probability that Yin wins on the third flip is the probability that the first flip is tails (1/2), the second flip is tails (1/2), AND the third flip is heads (1/2). This gives us (1/2) * (1/2) * (1/2) = 1/8.Total Probability of Yin Winning: We can see a pattern forming here. The probability that Yin wins on the nth flip is (1/2)^n. To find the total probability that Yin wins, we would sum up these probabilities for all flips from the second flip onwards.Sum of Infinite Geometric Series Formula: The sum of an infinite geometric series where the first term is a and the common ratio is r (|r| < 1) is given by the formula S = a / (1 - r). In this case, the first term is 1/4 (the probability Yin wins on the second flip) and the common ratio is 1/2 (because each subsequent term is half the previous term).Calculation of Yin's Total Winning Probability: Using the formula for the sum of an infinite geometric series, we get S = (1/4) / (1 - (1/2)) = (1/4) / (1/2) = (1/4) * 2 = 1/2. This means the total probability that Yin wins is 1/2.Fairness of Chosen Method: Since the probability that Corey wins is 1/2 and the probability that Yin wins is also 1/2, they both have an equal chance of getting to choose where they go for lunch. Therefore, the method they have chosen is fair.

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