인터랙티브 이항분포 시뮬레이션

이 실험 소개

A Galton board turns probability into a picture. Each ball falls through rows of pegs, and at every peg it bounces left or right by chance. The bin where it lands counts how many times it went right. One ball is unpredictable, but hundreds of balls build a reliable shape: the binomial distribution. Drop balls one at a time or a thousand at once, change the number of rows and the chance of bouncing right, and compare the pile with the formula and with the normal curve.

학습 목표

Describe a binomial experiment: a fixed number of independent trials with the same probability of success
Explain why there are C(n, k) routes to bin k and use this to find the probability of each bin
Calculate the mean np and the standard deviation √(np(1-p)) of a binomial distribution
See how a sample histogram gets closer to the expected distribution as the number of balls grows
Recognise when a binomial distribution is close to a normal distribution

사용 방법

1Drop 1 ball and follow its path. Count how many times it bounced right and check the bin number.
2Drop 10 balls, then 100. Turn on Show expected counts. How close are the bins to the blue marks?
3Add 1000 at once. Compare the sample mean and standard deviation with np and √(np(1-p)).
4Press New sample and repeat with the same settings. Which results change, and by how much?
5Set p to 0.2. Where does the peak move? Is the shape still symmetrical?
6Set n to 14 and p to 0.5 and turn on the normal approximation. Then try n = 3. When is the normal curve a good match?

모델

P (X = k) = (k n) p^{k} (1 - p)^{n - k}

Probability of landing in bin k after n rows

(k n) = \frac{n !}{k ! ( n - k )!}

Number of different routes with exactly k bounces to the right (a number in Pascal's triangle)

μ = n p, σ = n p (1 - p)

Mean and standard deviation of the binomial distribution

X \approx N (n p, n p (1 - p))

Normal approximation, good when np and n(1-p) are both at least about 5

가정

At each peg the ball goes right with the same probability p, independently of what it did at the pegs above.
Every ball meets exactly one peg per row and cannot skip a row or bounce sideways twice.
The random choices come from a seeded random number generator. The same sample number, n and p always give the same balls, so results can be repeated and discussed.
The animation is for illustration: the bin each ball lands in is decided by its list of left and right choices, not by a physics simulation of collisions.

수업에서 해 보기

Q1With 4 rows and p = 0.5, why is bin 2 six times as likely as bin 0?
Q2Why does the middle bin of a 14-row board hold only about a fifth of the balls, even though it is the most likely bin?
Q3A coin is tossed 10 times. What is the probability of exactly 5 heads? Use the board to check your answer.
Q4A multiple-choice test has 12 questions with 4 options each. A student guesses every answer. Which settings of the board model this, and what is the expected score?
Q5Why does the sample histogram look different each time you press New sample, while the blue expected counts stay the same?

자주 묻는 질문

Why do the bins form a bell shape?

There is only one route to each end bin, but many routes to the middle bins, because the left and right bounces can come in many different orders. Bins with more routes collect more balls, and the number of routes follows the rows of Pascal's triangle.

Is the distribution always symmetrical?

Only when p = 0.5. If p is smaller, the balls tend to stay left and the distribution is skewed to the right; the larger n is, the more symmetrical it becomes again.

What is the difference between the sample and the distribution?

The distribution gives the long-run share of balls expected in each bin. A sample is what actually happens with a limited number of balls, so it scatters around the expected values. The scatter shrinks in proportion as more balls are dropped.