1. What is Binomial Distribution Mean?

The mean of a binomial distribution represents the expected number of successes in a given number of independent trials, each with the same probability of success. It's a fundamental measure of central tendency for binomial experiments.

2. How Does the Calculator Work?

The calculator uses the binomial mean formula:

Where:

• \( n \) — Number of independent trials
• \( p \) — Probability of success on each trial
• \( \mu \) — Mean (expected value) of the distribution

Explanation: The formula shows that the expected number of successes grows linearly with both the number of trials and the probability of success.

3. Importance of Binomial Mean

Details: The mean helps predict outcomes in binomial experiments, quality control processes, medical trials, and any scenario with binary outcomes. It's essential for understanding the central tendency of discrete probability distributions.

4. Using the Calculator

Tips: Enter positive integer for number of trials (n) and probability (p) between 0 and 1. For example, for 10 coin flips (n=10) with fair coin (p=0.5), mean is 5 expected heads.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between mean and variance in binomial distribution?
A: While mean (μ = n×p) gives expected successes, variance (σ² = n×p×(1-p)) measures spread around this mean.

Q2: Can mean be non-integer?
A: Yes, while actual successes must be integers, the expected value (mean) can be fractional (e.g., 3.5 expected successes).

Q3: What if p > 1 or p < 0?
A: Probability must be between 0 and 1. The calculator validates this range.

Q4: How does increasing n affect the mean?
A: Mean increases linearly with n when p is held constant (direct proportionality).

Q5: When is binomial distribution symmetric?
A: When p = 0.5, the distribution is symmetric around its mean (n/2).