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

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%**.