An experiment consists of rolling a die and the tossing a coin once if the number on the die is even.If the number on the die is odd, the coin is tossed twice. Write the sample space for this experiment.

Explanation

To solve the problem, we need to determine the sample space for the given experiment, which consists of rolling a die and tossing a coin based on the outcome of the die. Let's break it down step by step:

### Step 1: Understand the outcomes of rolling a die
A standard die has six faces, numbered from 1 to 6. The outcomes are:
- Odd numbers: 1, 3, 5
- Even numbers: 2, 4, 6

### Step 2: Determine the coin toss based on the die outcome
- If the die shows an **even number** (2, 4, or 6), we toss the coin **once**.
- If the die shows an **odd number** (1, 3, or 5), we toss the coin **twice**.

### Step 3: List the outcomes for even numbers
For even numbers (2, 4, 6), the possible outcomes of tossing the coin once are:
- For 2: Heads (H), Tails (T) → Outcomes: 2H, 2T
- For 4: Heads (H), Tails (T) → Outcomes: 4H, 4T
- For 6: Heads (H), Tails (T) → Outcomes: 6H, 6T

So, the outcomes for even numbers are:
- 2H, 2T, 4H, 4T, 6H, 6T

### Step 4: List the outcomes for odd numbers
For odd numbers (1, 3, 5), the possible outcomes of tossing the coin twice are:
- For 1: HH, HT, TH, TT → Outcomes: 1HH, 1HT, 1TH, 1TT
- For 3: HH, HT, TH, TT → Outcomes: 3HH, 3HT, 3TH, 3TT
- For 5: HH, HT, TH, TT → Outcomes: 5HH, 5HT, 5TH, 5TT

So, the outcomes for odd numbers are:
- 1HH, 1HT, 1TH, 1TT, 3HH, 3HT, 3TH, 3TT, 5HH, 5HT, 5TH, 5TT

### Step 5: Combine the outcomes to form the sample space
Now, we combine the outcomes from both even and odd cases to form the complete sample space:
- Sample Space (S) = {2H, 2T, 4H, 4T, 6H, 6T, 1HH, 1HT, 1TH, 1TT, 3HH, 3HT, 3TH, 3TT, 5HH, 5HT, 5TH, 5TT}

### Final Sample Space
Thus, the complete sample space for the experiment is:
\[ S = \{ 2H, 2T, 4H, 4T, 6H, 6T, 1HH, 1HT, 1TH, 1TT, 3HH, 3HT, 3TH, 3TT, 5HH, 5HT, 5TH, 5TT \} \]

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