https://file.aiquickdraw.com/mathbot/file/1728017168386yk4gic5z.jpg

Let's go through the problem step by step based on the information in the image.

### (a) Write out the sample space.

When three fair coins are tossed, each coin can land either heads (H) or tails (T). The sample space is the set of all possible outcomes. Each outcome can be represented as a sequence of three letters where each letter is either H or T.

The sample space is:
\[ \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \} \]

### (b) Determine the probability of no tails.

"No tails" means all coins show heads. There's only one outcome that satisfies this condition: \( HHH \).

The total number of possible outcomes is 8 (since there are 2 possible results per coin and 3 coins: \( 2^3 = 8 \)).

The probability of no tails is:
\[ \frac{1 \text{ favorable outcome}}{8 \text{ possible outcomes}} = \frac{1}{8} \]

### (c) Determine the probability of at least one head.

To find the probability of getting at least one head, it is easier to first calculate the probability of the complementary event (getting no heads) and then subtract it from 1.

The only outcome with no heads is \( TTT \).

So, the probability of no heads (all tails) is:
\[ \frac{1 \text{ favorable outcome}}{8 \text{ possible outcomes}} = \frac{1}{8} \]

Therefore, the probability of at least one head is:
\[ 1 - \frac{1}{8} = \frac{7}{8} \]

### Summary:
- (a) The sample space is \( \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \} \).
- (b) The probability of no tails is \( \frac{1}{8} \).
- (c) The probability of at least one head is \( \frac{7}{8} \).

Would you like more details on any of these points or have any other questions?

Here are some related questions to expand on this topic:
1. What is the probability of getting exactly one head?
2. What is the probability of getting exactly two tails?
3. If one of the coins is biased and lands heads 60% of the time, how would the sample space change?
4. How would the probabilities change if we used four coins instead of three?
5. What is the probability of getting an equal number of heads and tails?

**Tip:** When calculating probabilities, sometimes it's easier to calculate the probability of the complementary event and subtract it from 1. This method is particularly useful when dealing with "at least" or "at most" probabilities.

Three fair coins are tossed. The two possible outcomes for a single coin are heads (H) and tails (T). Determine the sample space, the probability of no tails, and the probability of at least one head.

Explore how to determine the sample space when tossing three fair coins, the probability of no tails, and the probability of at least one head. Learn key concepts such as basic probability, combinatorics, and the complement rule in probability. Suitable for students in Grades 6-8.

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