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In probability, the sample space represents all possible outcomes of a random experiment. When three players each flip a fair coin, the sample space comprises all the combinations of heads (H) and tails (T) they can achieve in one toss. To visualize this, imagine each player flipping their coin simultaneously. The possible results are:

• HHH: All three show heads.
• HHT: First two show heads, third shows tail.
• HTH: First and third show heads, second shows tail.
• HTT: First shows head, last two show tails.
• THH: First shows tail, others show heads.
• THT: First and third show tails, second shows head.
• TTH: First two show tails, third shows head.
• TTT: All show tails.

Each of these represents a point in the sample space, illustrating all possible outcomes of the coin flips.

A fair coin is one that has an equal probability of landing on heads or tails when tossed. In mathematical terms, this means that for each toss, the probability of obtaining heads (H) is \( \frac{1}{2} \) and the same applies for tails (T), making it an unbiased coin. In the 'Odd Man Out' game, each player starts with a fair coin, implying every outcome within the sample space is initially equally likely.Using fair coins, no player has a statistical advantage over the others. Hence, each of the eight potential outcomes listed in the sample space has an equal probability of occurrence, specifically, \( \frac{1}{8} \) for each individual outcome.

A two-headed coin is a coin designed to show heads on both sides, ensuring the outcome is always heads (H) when flipped. In games involving probability, like 'Odd Man Out', a two-headed coin can significantly alter the outcomes. It introduces a bias where one result is guaranteed, skewing the fair chance characteristic necessary for an unbiased game. When a player opts for a two-headed coin in the game, their flips will always be heads. This reduces the sample space and alters the probability calculations because certain outcomes are no longer possible with its use. Specifically, only scenarios where other players' fair coins produce a tail alongside two heads (i.e., HHT and HTH) remain feasible for ending the game.

Calculating probability involves determining how likely an outcome is, given the sample space. For 'Odd Man Out', we want to find the probability of the game ending after the first coin flips.With all fair coins, we see 6 out of 8 outcomes result in an odd man out (HHT, HTH, THH, TTH, THT, HTT). Thus, the probability is \( \frac{6}{8} = \frac{3}{4} \). This means there's a 75% chance the game ends immediately.However, introduce a two-headed coin, and the odds change. The player guaranteed to show heads alters the sample space outcomes, leaving only HHT and HTH viable. Thus, the new probability calculation becomes \( \frac{2}{4} = \frac{1}{2} \), reducing the likelihood of concluding the game after a single toss to 50%.