1. Let's start by stating the problem: We want to understand the expected value and variance of a binomial distribution. 2. A binomial distribution models the number of successes in $n$ independent trials, each with a success probability $p$. The random variable $X$ representing the number of successes follows $X \sim \text{Binomial}(n, p)$. 3. The expected value or mean of $X$, denoted $E(X)$, tells us the average number of successes we expect after $n$ trials. It is given by: $$ E(X) = np $$ This means if you repeat the experiment many times, on average, $np$ successes will occur. 4. The variance of $X$, denoted $\text{Var}(X)$, measures how much the number of successes varies from the mean on average. It is given by: $$ \text{Var}(X) = np(1-p) $$ Here, $1-p$ is the probability of failure, so variance depends on both success and failure probabilities. 5. To summarize, for $X \sim \text{Binomial}(n,p)$: - Expected value: $$E(X) = np$$ - Variance: $$\text{Var}(X) = np(1-p)$$ These results are fundamental in probability and help predict outcomes and assess risk in binomial experiments.