You conduct the following experiment: You throw a fair six-sided die three times in a row and you note down the top number after every throw. Every throw is independent of the others. You combine the three numbers together to one element "(1st result, 2nd result, 3rd result)".

How many elements are in the sample space?

To determine the number of elements in the sample space for the given experiment, we need to consider the following:

- A fair six-sided die has 6 possible outcomes for each throw (1, 2, 3, 4, 5, 6).
- The die is thrown three times, and each throw is independent of the others.

For each throw, there are 6 possible outcomes. Since the throws are independent, the total number of possible outcomes for all three throws combined is given by multiplying the number of outcomes for each throw:

\[
6 \times 6 \times 6 = 6^3 = 216
\]

So, there are **216** elements in the sample space.

Would you like more details or have any questions? Here are some related questions for practice:

1. What is the probability of getting exactly one 6 in three rolls of a fair six-sided die?
2. How many possible outcomes include at least one 3 in three rolls of the die?
3. What is the probability of rolling a 6 on the first throw and a 1 on the second throw?
4. If you roll a die three times, what is the probability that all three rolls result in the same number?
5. How would the sample space change if the die was not fair but instead had different probabilities for each face?

**Tip:** When dealing with multiple independent events, remember that the total number of possible outcomes is the product of the number of outcomes for each individual event.

Understand how to calculate the sample space for independent dice rolls using a fair six-sided die. Learn about probability and the multiplication principle with clear examples. Suitable for Grades 7-9.

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