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A person plays a game of tossing a coin thrice. For each tail, he is given Rs.3 by the organiser of the game and for each head, he has to give Rs.2 to the organiser. Let X denote the amount gained or lost by the person. Show that X is a random variable and exhibit it as a function on the sample space of the experiment.

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To solve the problem, we need to define the random variable \( X \) based on the outcomes of tossing a coin three times. We will calculate the amount gained or lost for each possible outcome and show that \( X \) is a function of the sample space. ### Step 1: Define the Sample Space When a coin is tossed three times, the sample space \( S \) consists of all possible outcomes. Each outcome can be represented as a sequence of heads (H) and tails (T). The sample space is: \[ S = \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \} \] ...
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