One six-sided fair die is rolled, and one two-sided fair coin is tossed. If the coin turns up heads, then the number of spots showing on the die is the value (score) for that trial. If the coin is tails then twice the number of spots showing on the die is the value (score) for that trial. Determine the outcomes in the sample space and construct a distribution table. I figured out the sample space but I can't do the distribution table.

I'm doing this problem right now actually, and what i did was i made a chart of the scores, and then did the probability of each score, so like

1-1/12
2-1/6 and so on and so on

To construct the distribution table, we need to determine the probability of each outcome in the sample space. Let's first list the outcomes in the sample space:

- Coin: Heads, Die: 1
- Coin: Heads, Die: 2
- Coin: Heads, Die: 3
- Coin: Heads, Die: 4
- Coin: Heads, Die: 5
- Coin: Heads, Die: 6
- Coin: Tails, Die: 2
- Coin: Tails, Die: 4
- Coin: Tails, Die: 6
- Coin: Tails, Die: 8
- Coin: Tails, Die: 10
- Coin: Tails, Die: 12

Now, let's calculate the probability for each outcome. Since the die is fair and has six sides, each outcome has a 1/6 probability of occurring. The coin is also fair and has two sides, so the probability of getting heads is 1/2, and the probability of getting tails is also 1/2.

Here is the distribution table:

| Outcome | Probability |
|------------------------|-----------------------------|
| Coin: Heads, Die: 1 | (1/2) * (1/6) = 1/12 |
| Coin: Heads, Die: 2 | (1/2) * (1/6) = 1/12 |
| Coin: Heads, Die: 3 | (1/2) * (1/6) = 1/12 |
| Coin: Heads, Die: 4 | (1/2) * (1/6) = 1/12 |
| Coin: Heads, Die: 5 | (1/2) * (1/6) = 1/12 |
| Coin: Heads, Die: 6 | (1/2) * (1/6) = 1/12 |
| Coin: Tails, Die: 2 | (1/2) * (1/6) = 1/12 |
| Coin: Tails, Die: 4 | (1/2) * (1/6) = 1/12 |
| Coin: Tails, Die: 6 | (1/2) * (1/6) = 1/12 |
| Coin: Tails, Die: 8 | (1/2) * (1/6) = 1/12 |
| Coin: Tails, Die: 10 | (1/2) * (1/6) = 1/12 |
| Coin: Tails, Die: 12 | (1/2) * (1/6) = 1/12 |

In the distribution table, each outcome is listed with its corresponding probability.

To construct a distribution table, we need to list all the possible outcomes in the sample space and record the probability and value for each outcome.

In this scenario, we have a six-sided fair die and a two-sided fair coin. Let's start by listing all possible outcomes for each trial:

Die Outcome: 1, 2, 3, 4, 5, 6
Coin Outcome: Heads, Tails

To determine the complete sample space, we need to combine the outcomes of both trials. Since each outcome of the coin toss affects the value of the die roll, we have to account for each possibility:

For the heads outcome (H), the die outcome is the value (score):
- H1, H2, H3, H4, H5, H6

For the tails outcome (T), the value (score) is twice the die outcome:
- T2, T4, T6, T8, T10, T12

Now that we have the sample space, we can construct the distribution table. This table will show the probability and respective value for each outcome. Let's calculate the probability for each outcome:

Probability for heads outcome: 1/2 (since there are two equally likely outcomes for the coin)
Probability for tails outcome: 1/2 (since there are two equally likely outcomes for the coin)

Now let's construct the distribution table:

Outcome | Probability | Score
---------------------------------------
H1 | 1/2 | 1
H2 | 1/2 | 2
H3 | 1/2 | 3
H4 | 1/2 | 4
H5 | 1/2 | 5
H6 | 1/2 | 6
T2 | 1/2 | 2
T4 | 1/2 | 4
T6 | 1/2 | 6
T8 | 1/2 | 8
T10 | 1/2 | 10
T12 | 1/2 | 12

In the distribution table, we list each outcome, its corresponding probability, and the value (score) associated with it. We can see that for each outcome, the probability is 1/2, as we determined earlier.