Binomial Distribution Explorer

The controls, from top to bottom:

The bar chart shows $P(X = k)$ for every $k$ from 0 to $n$. Bars counted in your probability turn gold and a dashed line marks $k$. Two boxes give Mean μ = np and SD σ = √(np(1−p)) to four decimal places, and the Live probability box shows the answer, such as P(X ≥ 8) = 0.2277.

Switching the normal approximation on draws a teal normal curve with the same mean and SD over the bars, and a note under the probability. It reads Normal approximation OK when $np \ge 10$ and $n(1-p) \ge 10$, and Approximation unreliable otherwise. The curve is only a picture: the live probability is always the exact binomial value.

Watch the shape as you move $p$. At $p = 0.50$ the bars are symmetric; below 0.50 they are skewed right, above 0.50 skewed left. Raising $n$ makes any shape more symmetric. With the normal curve on, at $n = 40$, $p = 0.50$ it traces the bars closely; at $n = 10$, $p = 0.10$ it fits badly.

If $X$ counts the successes in $n$ trials with success probability $p$:

Problem: A basketball player makes 30% of her three-point attempts, and attempts are independent. In a practice she takes 20 three-pointers. Let $X$ = the number she makes. Find (a) $P(X = 6)$, (b) the probability she makes at least 8, and (c) the mean and standard deviation of $X$.

Step 1: Check the setting. Each shot is make or miss, shots are independent, $n = 20$ is fixed and $p = 0.30$ on every shot, so $X$ is binomial with $n = 20$, $p = 0.30$.

Step 2: Exactly 6. $P(X = 6) = \binom{20}{6}(0.30)^6(0.70)^{14} = 38{,}760 \times (0.30)^6 (0.70)^{14} \approx 0.1916$.

Step 3: At least 8. Use the complement: $P(X \ge 8) = 1 - P(X \le 7) = 1 - 0.7723 = 0.2277$.

Step 4: Mean and SD. $\mu_X = 20(0.30) = 6$ makes and $\sigma_X = \sqrt{20(0.30)(0.70)} = \sqrt{4.2} \approx 2.049$ makes. Over many 20-shot practices she averages 6 makes, and her count typically varies from 6 by about 2.05.

Check it in the simulator: set Trials to 20 and Success probability to 0.30. With Exact and $k = 6$, the box reads P(X = 6) = 0.1916. Choose At least and move $k$ to 8 for 0.2277. The mean and SD boxes show 6.0000 and 2.0494. Turn the normal approximation on: the note says it is unreliable because $np = 6.00 < 10$, and the bars are visibly right-skewed.

The binomial distribution is introduced in topic 2.10, and the probability formula, mean and standard deviation are on the AP Statistics formula sheet. Expect multiple-choice questions that ask for an exact or cumulative probability, and free-response parts that ask you to identify a binomial setting, calculate a probability with clear work, or interpret $\mu_X$ and $\sigma_X$ in context.