In this problem, we are asked to determine which outcomes are included in the sample space of tossing three coins, where the result is at most one tail.

Key conditions:

• "At most one tail" means there can be either zero tails or one tail in the outcome. Therefore, outcomes with more than one tail are excluded.

Step 1: Analyzing the sample space (outcomes with at most one tail)

We need to select the outcomes from the list that satisfy this condition. Let’s go through each one:

• TTT: This outcome has three tails, so it is excluded.
• TTH: This outcome has two tails, so it is excluded.
• HTH: This outcome has two tails, so it is excluded.
• THH: This outcome has one tail, so it is included.
• THT: This outcome has two tails, so it is excluded.
• HTT: This outcome has two tails, so it is excluded.
• HHH: This outcome has zero tails, so it is included.
• HHT: This outcome has one tail, so it is included.

Step 2: Listing the elements that meet the condition

Based on the condition that there is at most one tail, the elements of the event are:

• THH
• HHH
• HHT

Final Answer:

The following sets of elements are included in the sample space:

• THH
• HHH
• HHT

Would you like more details or have any further questions?

Here are 5 related questions:

1. How would the sample space change if the condition was "at least two tails"?
2. What is the probability of getting exactly one tail in this scenario?
3. How many total outcomes are possible when tossing three coins?
4. If the coins were not distinguishable, how would the problem be different?
5. How would you calculate the probability of getting no tails?

Tip: When analyzing probability problems, always check how conditions like "at most" or "at least" restrict the outcomes!