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First, we need to create a sample space of the possible outcomes when a coin is tossed twice. There are 4 possible outcomes in this case: 1. First toss is a head, and the second toss is a head (HH) 2. First toss is a head, and the second toss is a tail (HT) 3. First toss is a tail, and the second toss is a head (TH) 4. First toss is a tail, and the second toss is a tail (TT)

Now, we need to create the list of possible values for \(X\), the number of tails that come up when a coin is tossed twice. We can examine the 4 possible outcomes to determine the value of \(X\) for each: 1. HH: Number of tails = 0 (X = 0) 2. HT: Number of tails = 1 (X = 1) 3. TH: Number of tails = 1 (X = 1) 4. TT: Number of tails = 2 (X = 2) From this analysis, we can see that the possible values for \(X\) are 0, 1, and 2.

In order to calculate the probabilities for each possible value of \(X\), we need to count the number of occurrences of each \(X\) value in the sample space and divide by the total number of outcomes in the sample space. The total number of outcomes in this case is 4 (HH, HT, TH, TT). 1. Probability of \(X\) = 0 (no tails coming up): There is only one outcome that corresponds to \(X\) = 0 (HH). So, the probability is: \(P(X=0) = \frac{1}{4}\) 2. Probability of \(X\) = 1 (one tail coming up): There are two outcomes that correspond to \(X\) = 1 (HT and TH). So, the probability is: \(P(X=1) = \frac{2}{4} = \frac{1}{2}\) 3. Probability of \(X\) = 2 (two tails coming up): There is only one outcome that corresponds to \(X\) = 2 (TT). So, the probability is: \(P(X=2) = \frac{1}{4}\)

The possible values for \(X\) and their respective probabilities are: 1. \(P(X=0) = \frac{1}{4}\) 2. \(P(X=1) = \frac{1}{2}\) 3. \(P(X=2) = \frac{1}{4}\) Now, the student should understand the possible values for the random variable \(X\) and how to calculate their respective probabilities when a coin is tossed twice.