16. You toss a coin four times. What is the probability of tossing tails exactly half of the time?

To find the probability of tossing tails exactly half of the time when tossing a coin four times, we can follow these steps: 1. **Identify the total number of tosses**: We are tossing the coin \( n = 4 \) times. 2. **Determine the number of successful outcomes**: We want to find the probability of getting tails exactly \( k = 2 \) times (which is half of 4). 3. **Use the binomial probability formula**: The probability of getting exactly \( k \) successes (tails) in \( n \) trials (tosses) is given by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where: - \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes in \( n \) trials. - \( p \) is the probability of success on a single trial (for a fair coin, \( p = 0.5 \)). - \( (1-p) \) is the probability of failure. 4. **Calculate the binomial coefficient**: \[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \] 5. **Substitute values into the formula**: \[ P(X = 2) = \binom{4}{2} (0.5)^2 (0.5)^{4-2} = 6 \times (0.5)^2 \times (0.5)^2 = 6 \times (0.5)^4 \] 6. **Calculate \( (0.5)^4 \)**: \[ (0.5)^4 = \frac{1}{16} \] 7. **Final calculation**: \[ P(X = 2) = 6 \times \frac{1}{16} = \frac{6}{16} = \frac{3}{8} \] Thus, the probability of tossing tails exactly half of the time (2 tails in 4 tosses) is \( \frac{3}{8} \).