The Binomial Distribution
If each of $n$ independent trials has success probability $p$, the number of successes $X$ follows a binomial distribution: $P(X = k) = \dbinom{n}{k} p^k (1-p)^{n-k}$. Its mean is $\mu = np$ and its standard deviation $\sigma = \sqrt{np(1-p)}$. The red bars are the region you selected; the probability is the sum of their heights.
What to notice
- "At least $k$" and "more than $k$" are not the same: $P(X \ge 4) = 1 - P(X \le 3)$. This is the most common exam mistake; look at the bars to see which one is included.
- With $p = 0.5$ the distribution is symmetric; as $p$ gets smaller it becomes right-skewed, as it gets larger left-skewed.
- Turn on the normal curve and increase $n$: once $np \ge 10$ and $n(1-p) \ge 10$ the bars sit on the curve. In a case like $p = 0.05$, $n = 40$ they do not.
- When $n$ quadruples the mean quadruples, but the standard deviation only doubles.