Describe in words (do not list all elements) the sample space of each experiment, find how many elements it contains, and state one of the elements: Rolling a 6-sided die, then flipping that many coins.

The experiment involves two stages: first, rolling a standard 6-sided die and then flipping a number of coins equal to the result of the die roll.

The sample space consists of all possible outcomes of this two-step process. In the first step, the die can land on any of the numbers from 1 to 6. Depending on the number rolled, the second step involves flipping that many coins. If the die shows 1, you flip 1 coin, and the outcome is either heads or tails. If the die shows 2, you flip 2 coins, and each coin can independently land on heads or tails, giving you multiple combinations of heads and tails for that round. Similarly, for any die result, the number of coin flips corresponds to the die roll, and the outcomes are all the possible combinations of heads and tails for that many flips.

The total number of elements in the sample space is calculated by summing the number of possible outcomes for each die result:
- If you roll a 1, there are 2 possible outcomes (since 1 coin has 2 outcomes).
- If you roll a 2, there are 4 possible outcomes (since 2 coins have \(2^2 = 4\) combinations).
- If you roll a 3, there are 8 possible outcomes (since 3 coins have \(2^3 = 8\) combinations).
- If you roll a 4, there are 16 possible outcomes (since 4 coins have \(2^4 = 16\) combinations).
- If you roll a 5, there are 32 possible outcomes (since 5 coins have \(2^5 = 32\) combinations).
- If you roll a 6, there are 64 possible outcomes (since 6 coins have \(2^6 = 64\) combinations).

Thus, the total number of elements in the sample space is \( 2 + 4 + 8 + 16 + 32 + 64 = 126 \) possible outcomes.

One of the elements in this sample space could be: "Roll a 3, flip the coins, and get heads, tails, heads."

Would you like further details on this? Here are 5 related questions you could explore:
1. How would the sample space change if you used an 8-sided die?
2. What is the probability of getting heads on all coin flips when rolling a 4 on the die?
3. How does the sample space size grow as you increase the number of sides on the die?
4. If the die is biased, how would that affect the probabilities of different outcomes?
5. How can you model this experiment using a probability tree diagram?

**Tip:** When calculating total outcomes in multi-step experiments, consider each step independently and then multiply or sum the outcomes based on the structure of the process.

Describe in words the sample space of each experiment, find how many elements it contains, and state one of the elements: Rolling a 6-sided die, then flipping that many coins.

Explore the sample space and calculate how many elements it contains when rolling a 6-sided die and flipping coins equal to the die's result. Understand how to compute the total outcomes using combinatorics, and get a clear explanation of each step in this probability experiment. Suitable for Grades 9-11.

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