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

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UpStudy AI · Accepted Answer

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 $$ \frac{3}{8} $$.
The probability of tossing tails exactly half of the time in four coin tosses is $$ \frac{3}{8} $$.

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

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