Two dice are rolled and two fair coins are tossed. Let X be the sum of the number of spots that show on the top faces of the dice and the number of coins that land heads up. The expected value of X is _____ .

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To find the expected value of X, we need to calculate the probabilities and expected values for each possible outcome.

We have two dice, each having six sides and numbered from 1 to 6. So, the sum of the numbers on the top faces of the dice can be any number from 2 to 12.

Similarly, we have two fair coins, each with two possible outcomes - heads (H) or tails (T). So, the number of coins that land heads up can be 0, 1, or 2.

To find the probabilities for each outcome, we need to consider the total number of possible outcomes and the number of favorable outcomes for each event.

1. The sum of the numbers on the dice:
- There are 6 possible outcomes for each dice, so the total number of possible outcomes for both dice is 6 * 6 = 36.
- The number of favorable outcomes for each sum is as follows:
- For a sum of 2, there is only 1 outcome: (1, 1).
- For a sum of 3, there are 2 outcomes: (1, 2) and (2, 1).
- For a sum of 4, there are 3 outcomes: (1, 3), (2, 2), and (3, 1).
- This pattern continues until a sum of 7, where there are 6 outcomes.
- After that, the number of outcomes decreases symmetrically until a sum of 12, which has only 1 outcome: (6, 6).
- So, the total number of favorable outcomes for each sum is given by the sequence: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1.

2. The number of coins that land heads up:
- For each coin, there are 2 possible outcomes (H or T), so the total number of possible outcomes for both coins is 2 * 2 = 4.
- The number of favorable outcomes for each number of heads is as follows:
- 0 heads: 1 outcome (T, T)
- 1 head: 2 outcomes (H, T) and (T, H)
- 2 heads: 1 outcome (H, H)

Now, we can calculate the probabilities for each outcome:

Probability of a sum and number of heads:
- P(X = 2) = 1/36 * 1/4 = 1/144
- P(X = 3) = 2/36 * 2/4 = 4/144
- P(X = 4) = 3/36 * 2/4 = 6/144
- P(X = 5) = 4/36 * 2/4 = 8/144
- P(X = 6) = 5/36 * 2/4 = 10/144
- P(X = 7) = 6/36 * 2/4 = 12/144
- P(X = 8) = 5/36 * 2/4 = 10/144
- P(X = 9) = 4/36 * 2/4 = 8/144
- P(X = 10) = 3/36 * 2/4 = 6/144
- P(X = 11) = 2/36 * 2/4 = 4/144
- P(X = 12) = 1/36 * 1/4 = 1/144

Now, we can calculate the expected value of X by multiplying each outcome by its respective probability and summing them up:

E(X) = 2 * (1/144) + 3 * (4/144) + 4 * (6/144) + 5 * (8/144) + 6 * (10/144) + 7 * (12/144) + 8 * (10/144) + 9 * (8/144) + 10 * (6/144) + 11 * (4/144) + 12 * (1/144)

Simplifying this expression, we find:

E(X) = 7

Therefore, the expected value of X is 7.