.Calculate the probability of obtaining 2 heads in 2 tosses of a coin ,use tree diagram ,calculate the probability of obtaining a 2 followed by a 5 in 2 draws of a die use tree diagrams and using proper formulas 

Let's break down both problems, using tree diagrams and proper probability formulas.

### 1. Probability of Obtaining 2 Heads in 2 Tosses of a Coin

#### **Tree Diagram for Coin Tosses**

- **First Toss:** Two possible outcomes: Head (H) or Tail (T).
- **Second Toss:** For each outcome of the first toss, there are again two possible outcomes: Head (H) or Tail (T).

The tree diagram is:

```
       Start
        / \
      H     T
     / \   / \
    H   T H   T
```

#### **Outcomes and Probabilities**
- **HH**: Head on the first toss and Head on the second toss.
- **HT**: Head on the first toss and Tail on the second toss.
- **TH**: Tail on the first toss and Head on the second toss.
- **TT**: Tail on the first toss and Tail on the second toss.

Each outcome has a probability of \(\frac{1}{2}\) for each toss. So:

- \(P(\text{H on 1st toss}) = \frac{1}{2}\)
- \(P(\text{H on 2nd toss}) = \frac{1}{2}\)

The probability of both events happening (2 heads) is:

\[
P(\text{HH}) = P(\text{H on 1st toss}) \times P(\text{H on 2nd toss}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]

### 2. Probability of Obtaining a 2 Followed by a 5 in 2 Draws of a Die

#### **Tree Diagram for Die Rolls**

- **First Roll:** Six possible outcomes: 1, 2, 3, 4, 5, 6.
- **Second Roll:** For each outcome of the first roll, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

The tree diagram is more extensive, but here are the relevant branches:

```
       Start
        / | \ ...
      1   2   3 ... 
     /|   /|  /|
    1 2  1 2 1 2 ...
         |
         5
```

#### **Outcomes and Probabilities**
- The outcome of interest is rolling a 2 followed by a 5: (2, 5).

Each roll of the die has a probability of \(\frac{1}{6}\) for any specific number. So:

- \(P(\text{2 on 1st roll}) = \frac{1}{6}\)
- \(P(\text{5 on 2nd roll}) = \frac{1}{6}\)

The probability of both events happening (rolling a 2 followed by a 5) is:

\[
P(2 \text{ followed by } 5) = P(\text{2 on 1st roll}) \times P(\text{5 on 2nd roll}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}
\]

### Summary of Results:
1. **Probability of obtaining 2 heads in 2 tosses of a coin:** \(\frac{1}{4}\).
2. **Probability of obtaining a 2 followed by a 5 in 2 draws of a die:** \(\frac{1}{36}\).

Would you like to explore any related questions or need further clarification?

Here are some related questions to explore:

1. What is the probability of getting at least one head in 2 tosses of a coin?
2. How does the probability change if the coin is biased?
3. What is the probability of getting a 2 followed by a 2 in 2 draws of a die?
4. How would the probability change if there were 3 coin tosses?
5. What is the probability of getting a tail followed by a tail in 2 coin tosses?

**Tip:** For sequential events, multiplying the probabilities of individual independent events helps you calculate the probability of the combined event.

Learn how to calculate the probability of specific sequences in coin tosses and die rolls using tree diagrams and probability formulas. Understand the likelihood of obtaining 2 heads in 2 coin tosses and a 2 followed by a 5 in 2 die rolls.

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