In a binomial distribution, the parameter \( p \) represents the probability of success in a single trial. 

The binomial distribution is defined as \( \text{Binomial}(n, p) \), where:

- \( n \) is the number of independent trials.
- \( p \) is the probability of success on each trial.

The probability mass function (PMF) of a binomial distribution is given by:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( k \) is the number of successes,
- \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials.

### Key Points:
1. **Range of \( p \)**: \( 0 \leq p \leq 1 \).
2. **Interpretation**: If \( p = 0.5 \), each trial is equally likely to result in success or failure.
3. **Expected Value**: The mean or expected value of the binomial distribution is \( \mu = np \).
4. **Variance**: The variance of the binomial distribution is \( \sigma^2 = np(1 - p) \).
5. **Example**: If you flip a fair coin 10 times (where \( p = 0.5 \) for heads), the number of heads follows a \( \text{Binomial}(10, 0.5) \) distribution.

Would you like further details or have any questions? Here are some related topics you might find interesting:

1. How is the binomial distribution different from the normal distribution?
2. What is the cumulative distribution function (CDF) of a binomial distribution?
3. How does the value of \( p \) affect the shape of the binomial distribution?
4. How can we approximate the binomial distribution using the normal distribution?
5. What are some real-world examples of binomial distributions?

**Tip**: In a binomial distribution, if \( p \) is close to 0 or 1, the distribution is skewed, while if \( p \) is around 0.5, the distribution is more symmetric.

Learn about the probability parameter p in binomial distributions. This page covers key concepts, such as the binomial formula, expected value, and variance, suitable for high school students in Grades 9-12.

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