Rework problem 29 from section 1.4 of your text, involving the flipping of a coin. A coin is flipped. If a heads is flipped, then the coin is flipped 5 more times and the number of heads flipped is noted; otherwise (i.e., a tails is flipped on the initial flip), then the coin is flipped 4 more times and the result of each flip (i.e., heads or tails) is noted successively. How many possible outcomes are in the sample space of this experiment?

To solve this problem, we need to determine the number of possible outcomes in the sample space for the coin-flipping experiment.

### Step-by-Step Breakdown:

1. **Initial Flip**:
   - The first flip determines two branches: either a **heads (H)** or a **tails (T)**.
   
2. **If Heads is Flipped on the Initial Flip**:
   - The coin is flipped **5 more times**, and we are interested in the **number of heads** flipped in these 5 flips.
   - Each flip can result in either heads or tails, so there are \( 2^5 = 32 \) possible outcomes for the 5 flips following the initial heads.
   
3. **If Tails is Flipped on the Initial Flip**:
   - The coin is flipped **4 more times**, and we record the result of **each individual flip**.
   - Since each flip has 2 possible outcomes (heads or tails), the number of possible outcomes for these 4 flips is \( 2^4 = 16 \).

4. **Total Sample Space**:
   - The initial flip has 2 possible outcomes (heads or tails).
   - If the initial flip is heads, there are 32 possible outcomes.
   - If the initial flip is tails, there are 16 possible outcomes.
   
   Therefore, the total number of possible outcomes in the sample space is:
   \[
   2 \times (32 + 16) = 2 \times 48 = 96
   \]

Thus, the total number of possible outcomes in the sample space of this experiment is **96**.

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Would you like further explanation or details on any part of this solution? 

Here are five related questions to deepen your understanding:

1. How would the solution change if we flipped the coin 6 times instead of 5 after the initial heads?
2. What if the coin used was biased, and heads had a different probability than tails?
3. How many outcomes would there be if both the heads and tails paths involved 5 additional flips?
4. What is the probability of getting exactly 3 heads in the 5 additional flips after flipping heads initially?
5. How does this problem relate to binomial distributions?

**Tip**: When solving probability problems, always carefully consider the different branches or stages of the experiment. Each branch multiplies the number of possible outcomes.

Explore a coin-flipping experiment where the number of heads or tails is recorded based on the first flip. Learn how to calculate the total number of possible outcomes in the sample space using probability concepts. Suitable for high school students in Grades 10-12.

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