1. **State the problem:** We want to find the expected value and variance of a binomial distribution. 2. **What is a binomial distribution?** It describes the number of successes in $n$ independent trials, each with a probability $p$ of success. 3. **Expected value (mean) of binomial distribution:** The expected value $E(X)$ is the average number of successes in $n$ trials. 4. The formula for the expected value is $$E(X) = np$$ where $n$ is the number of trials and $p$ is the probability of success in each trial. 5. **Variance of binomial distribution:** The variance measures the spread of the distribution around the mean. 6. The formula for variance is $$Var(X) = np(1-p)$$ where $1-p$ is the probability of failure in each trial. 7. **Summary:** For a binomial random variable $X \sim Binomial(n,p)$, - Expected value: $$E(X) = np$$ - Variance: $$Var(X) = np(1-p)$$ This means on average you expect $np$ successes, and the variability is quantified by $np(1-p)$.