The normal distribution

Probability as area: the 68-95-99.7 rule, what σ does, and how random draws settle onto the exact answer. About 9 minutes, A-level Statistics.

What you will do

1. Probability is area. This is the standard normal curve, mean = 0 and standard deviation = 1. The shaded area between the red bounds is the probability that a random value lands in that range. Right now it covers -1 to 1 and the engine says it is prob.
2. Two standard deviations. Drag the red lines out to -2 and 2 and watch the probability climb.
3. What does σ do?. Now = 2: the curve is twice as wide and half as tall — total area is still 1. The bounds stay at 1, so they now capture less.
4. Roll the dice. Probability is also what you see in the long run. Simulate at least 100 random draws and compare the fraction inside the bounds with the exact probability.
5. Counting successes. A binomial distribution counts successes in n independent trials, here 20 fair coin flips. The bars are exact probabilities, and the highlighted ones are the whole numbers from 8 to 12.
6. Your turn. Everything is unlocked. Change the distribution and its parameters, move the bounds, and compare simulated draws with the exact answer.