### Part 1: Probability of Spinning 2 Prime Numbers
To find the probability of spinning 2 prime numbers, we first need to determine the probability of spinning a single prime number and then use that to find the probability of spinning two primes in a row.
1. Identify Prime Numbers: We have a list of prime numbers within the context: {2, 3, 5, 7, 11}.
2. Total Numbers: Assuming the spinner has 12 numbers (likely 1 through 12).
3. Count of Prime Numbers: There are 5 prime numbers (2, 3, 5, 7, 11).
Next, we calculate the probability of spinning a prime number on any given spin:
[tex]\[ \text{Probability of spinning a prime number} = \frac{\text{Number of prime numbers}}{\text{Total numbers}} = \frac{5}{12} \approx 0.4167 \][/tex]
Finally, to find the probability of spinning two prime numbers consecutively:
[tex]\[ \text{Probability of spinning two prime numbers} = \left( \frac{5}{12} \right) \times \left( \frac{5}{12} \right) = \left( 0.4167 \right)^2 \approx 0.1736 \][/tex]
Therefore, the probability of spinning two prime numbers is approximately [tex]\(0.1736\)[/tex].
### Part 2: Coin Toss Outcomes
When tossing a coin two times, we need to list all possible outcomes. Each toss has two possible results: Heads (H) or Tails (T).
Let's systematically list all combinations:
- First Toss: Heads (H)
- Second Toss: Heads (H) ⇒ (H, H)
- Second Toss: Tails (T) ⇒ (H, T)
- First Toss: Tails (T)
- Second Toss: Heads (H) ⇒ (T, H)
- Second Toss: Tails (T) ⇒ (T, T)
Thus, we have the following possible outcomes:
1. (H, H)
2. (H, T)
3. (T, H)
4. (T, T)
Table of Outcomes:
| First Toss | Second Toss | Outcome |
|------------|-------------|---------|
| H | H | (H, H) |
| H | T | (H, T) |
| T | H | (T, H) |
| T | T | (T, T) |
Or using a tree diagram:
1. Start with the first toss (two branches: H and T).
2. From each branch of the first toss, create two more branches for the second toss (each splitting into H and T).
```
Start
/ \
H T
/ \ / \
H T H T
(H,H)(H,T)(T,H)(T,T)
```
