21. Suppose I flip a fair coin 3 times. What is the probability that I get tails exactly twice?

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UpStudy AI · Accepted Answer

To determine the probability of getting **exactly two tails** when flipping a fair coin **three times**, you can use the principles of probability and combinations. Here's a step-by-step explanation:

### 1. Understand the Basics

- **Fair Coin:** This means that the probability of getting **Heads (H)** is equal to the probability of getting **Tails (T)**, which is 0.5 each.
- **Number of Flips:** 3

### 2. Total Possible Outcomes

When you flip a coin three times, each flip has 2 possible outcomes (H or T). Therefore, the total number of possible outcomes is:

$$
2^3 = 8
$$

These outcomes are:
1. HHH
2. HHT
3. HTH
4. HTT
5. THH
6. THT
7. TTH
8. TTT

### 3. Favorable Outcomes

We want exactly two tails. Let's identify which of the above outcomes meet this criterion:

1. **HHT** — Two tails
2. **HTT** — Two tails
3. **THT** — Two tails
4. **TTH** — Two tails

Wait, there are **4** outcomes with exactly two tails. It seems there was a miscount in the initial explanation. Let's correct that.

**Correction:**
There are actually **3** favorable outcomes:
1. **HTT**
2. **THT**
3. **TTH**

**Note:** The outcome **HHT** has only one tail, and **THH** has one tail as well. **TTT** has three tails.

### 4. Calculating the Probability

The probability $$ P $$ of an event is given by:

$$
P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}
$$

Plugging in the numbers:

$$
P = \frac{3}{8} = 0.375
$$

So, the probability is **$$\frac{3}{8}$$** or **37.5%**.

### 5. Using Combinations

Alternatively, you can use combinations to calculate the probability:

$$
P = \binom{n}{k} \times (p)^k \times (1-p)^{n-k}
$$

Where:
- $$ n = 3 $$ (number of trials)
- $$ k = 2 $$ (number of desired successes, i.e., tails)
- $$ p = 0.5 $$ (probability of getting a tail on a single flip)

Calculating the combination:

$$
\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3
$$

Now, plug the values in:

$$
P = 3 \times (0.5)^2 \times (0.5)^{1} = 3 \times 0.25 \times 0.5 = 0.375
$$

Again, the probability is **$$\frac{3}{8}$$** or **37.5%**.

### **Conclusion**

The probability of getting exactly two tails in three flips of a fair coin is **$$\frac{3}{8}$$** or **37.5%**.
The probability of getting exactly two tails in three flips of a fair coin is 3/8 or 37.5%.

Suppose I flip a fair coin 3 times. What is the probability that I get tails exactly twice?

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Solución

1. Understand the Basics

2. Total Possible Outcomes

3. Favorable Outcomes

4. Calculating the Probability

5. Using Combinations

Conclusion

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