A fair die is rolled and a fair coin is flipped. The 12 outcomes in the sample space are equally likely. Define the random variable as follows: if the coin shows heads and the die shows a number greater than 4; otherwise . Which table correctly represents the theoretical probability distribution of ?
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Explanation: This problem involves developing theoretical probability distributions and calculating expected values for discrete random variables. In this model, probabilities are derived from the 12 equally likely outcomes of a fair die roll and coin flip, each with probability 1/12. Each outcome maps to X=1 if the coin is heads and die >4 (i.e., heads with 5 or 6), otherwise X=0. The probability P(X=x) is determined by counting favorable outcomes: P(X=1)=2/12=1/6, P(X=0)=10/12=5/6. The expected value E(X) is computed as 1*(1/6) + 0*(5/6) = 1/6. A common misconception is that all X-values are equally likely, but X=0 is far more probable due to the conditions. To solve similar problems, list all outcomes, map each to its X, and combine probabilities for each distinct x.