To find the probability that exactly 50 out of 100 coin flips result in heads, we need to make use of the binomial distribution. Here are the steps:
1. Understanding the problem:
- Total number of flips (n) = 100
- Desired number of heads (k) = 50
- Probability of getting heads in a single flip (p) = 0.5 (since it’s a fair coin)
2. Defining the binomial distribution:
- The binomial distribution describes the number of successes (in this case, heads) in a fixed number of independent trials (coin flips), with each trial having the same probability of success.
3. Probability Mass Function (PMF):
- The probability of getting exactly [tex]\( k \)[/tex] successes (heads) in [tex]\( n \)[/tex] trials (flips) is given by the binomial probability formula:
[tex]\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\][/tex]
Here:
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, also known as "n choose k" or [tex]\(\frac{n!}{k! (n - k)!}\)[/tex], which counts the number of ways to choose k successes out of n trials.
- [tex]\( p^k \)[/tex] is the probability of getting heads k times.
- [tex]\( (1 - p)^{n - k} \)[/tex] is the probability of getting tails the remaining times.
4. Applying the numbers:
- Substituting [tex]\( n = 100 \)[/tex], [tex]\( k = 50 \)[/tex], and [tex]\( p = 0.5 \)[/tex] into the binomial probability formula.
[tex]\[
P(X = 50) = \binom{100}{50} (0.5)^{50} (0.5)^{50}
= \binom{100}{50} (0.5)^{100}
\][/tex]
5. Calculating the binomial coefficient:
- The binomial coefficient [tex]\(\binom{100}{50}\)[/tex] represents the number of ways to choose 50 flips out of 100, where order does not matter.
6. Combining the elements:
- Once you have the binomial coefficient and the probability terms, you multiply them together to get the final probability.
After performing these calculations (considering high computational values and steps), the result is:
[tex]\[ P(X = 50) \approx 0.07958923738717875 \][/tex]
Therefore, the probability that exactly 50 of the 100 coin flips were heads is approximately 0.0796 or about 7.96%.
