Binomial & Poisson on different data: survival & quakes¶

Part 1 · Lesson 8 · another dataset · data: Titanic · USGS earthquakes

↩ A companion to Lesson 8: Binomial & Poisson — counting events.

↩ Back: Lesson 8: Binomial & Poisson — counting events · ↑ Course map · Also: The code behind it →

The core lesson met the two great counting models — the Binomial (how many successes in a fixed number of tries) and the Poisson (how many rare events in a fixed window of time) — on low-birthweight babies and busy worldwide earthquakes. Here we meet the same two models on different data, chosen on purpose to look different:

• a Binomial with a much bigger success chance — survival aboard the Titanic (about 39 in 100, versus the births' 8 in 100), so its histogram sits further from the wall at zero;
• a Poisson that is rarer — only the larger magnitude-5+ earthquakes per day (about 3.6 a day, versus the core's busy 17), so its shape is lopsided and choppy instead of a near-symmetric mound.

Same two formulas, different rates — and watching them bend with the rate is the whole point. Run each grey cell with Shift+Enter, then tinker.

New to notebooks? The grey boxes below are code. Click one and press Shift + Enter to run it — the result (a number, a table, or a chart) appears right underneath. Run them top to bottom. You do not need to understand the first "Setup" cell — it just loads the tools and the data, so run it and move on. And you can't break anything: re-running a cell is always safe. (Brand new to all this? The five-minute Lesson 0 — How these notebooks work shows you the basics first.)

1) The building block: a Bernoulli trial¶

Every count in this lesson is built from the simplest random thing there is: a single yes/no event. A trial with exactly two outcomes — "success" (1) or "failure" (0) — is a Bernoulli trial, named after Jacob Bernoulli.

• The math. One number does the whole job: $p$, the probability of success. Then $P(\text{success}) = p$ and $P(\text{failure}) = 1 - p$.
• The meaning. $p$ is the long-run fraction of trials that come up "success."
• The interpretation. For us, one passenger either survived the sinking (1) or did not (0). The Survived column is already a column of 0s and 1s — a stack of Bernoulli trials we can read straight off the data. ("Success" is just statistics jargon for "the thing we're counting"; here it happens to be a genuinely hopeful one.)

Let's read $p$, the survival rate, straight off the data.

Contrast with the core lesson. There, "success" (a low-birthweight baby) had $p \approx 0.08$ — a fairly rare outcome. Here $p \approx 0.39$ is much closer to a coin flip. That single difference will visibly reshape everything below: a bigger $p$ pushes the whole distribution to the right, away from the wall at zero. (The women-vs-men split is a strong reminder that this one overall $p$ averages over very different groups — a thread the probability lessons pull on.)

2) The Binomial distribution — successes in $n$ tries¶

Now stack $n$ independent Bernoulli trials (independent = one trial's result tells you nothing about the next), each with the same $p$, and count the successes. That count is a Binomial random variable, written $X \sim \text{Binomial}(n, p)$ — read the squiggle "$\sim$" as "follows the … distribution" (so: "$X$ follows a Binomial distribution with settings $n$ and $p$").

Picture a random group of $n = 20$ passengers, each drawn from the full ship's-worth of people. How many will turn out to be survivors? It could be 0, or 1, ... up to 20 — but not all answers are equally likely.

• The math. The probability of exactly $k$ successes is $$P(X = k) = \binom{n}{k}\, p^{k}\,(1-p)^{\,n-k}.$$ The piece $p^k(1-p)^{n-k}$ is the chance of one specific pattern with $k$ successes; $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$ counts how many different patterns give $k$ successes.
• The meaning. It spreads a total probability of 1 across the possible counts $0,1,\dots,n$ — this list is the PMF (probability mass function).
• The interpretation. It tells us, before we look, how surprising any particular number of survivors in a group of 20 would be.

Two summary numbers fall out of the formula:

$$\text{mean} = np, \qquad \text{variance} = np(1-p).$$

Note the center. The mean $np \approx 7.7$ survivors out of 20 sits well to the right — about 39% of the group, as it must. In the core lesson the rare-event mean was barely 2 out of 30, pinned near the left wall. Bigger $p$, center shifts right: that is the lever we are watching.

3) Simulate it — don't just believe the formula¶

Here is the heart of "by seeing it happen." Instead of trusting the formula, we'll run the experiment thousands of times: draw a random group of 20 passengers (with replacement — we treat the ship's roster as an endless pool with the same survival chance $p$, so each draw is a clean Bernoulli trial), count the survivors, and repeat. Pile the counts into a histogram and the Binomial's shape appears on its own.

The tinkerable knobs are at the top of the cell.

Read those two lines. The simulated mean and variance land right on the formulas $np$ and $np(1-p)$ — we never plugged the formulas in; the experiment produced them. Now overlay the exact PMF on the histogram of simulated counts. (The bars are scaled to probabilities — their heights sum to 1 — so they line up directly with the formula's red dots, instead of showing raw counts in the thousands.)

The interpretation. The red dots sit right on the blue bars: formula and experiment agree. In a random group of 20 passengers we'd most often see 7 or 8 survivors, wouldn't blink at 5 or 10, and either extreme — very few or most of the group surviving — is genuinely uncommon. Because $p$ is near a half, this mound is fairly symmetric and sits in the middle of the range, not crushed against zero the way the rare low-birthweight count was.

A sanity check you already trust: a fair coin¶

To see that the Binomial isn't special to Titanic data, set $p = 0.5$ and flip 20 coins. The shape becomes a clean symmetric mound centered on 10 heads — and notice how close our survival mound already was to it, because $0.39$ isn't far from $0.5$.

4) The Poisson distribution — rare events per interval¶

The Binomial needs a fixed number of trials $n$. But many real questions have no obvious $n$: how many earthquakes today? How many emails this hour? There were countless moments when a quake could have struck, each with a tiny probability. For "lots of chances, each rare," we use the Poisson distribution, written $X \sim \text{Poisson}(\lambda)$.

• The math. With a single rate $\lambda$ (the average number of events per interval), $$P(X = k) = \frac{e^{-\lambda}\,\lambda^{k}}{k!}, \qquad k = 0, 1, 2, \dots$$
• The meaning. $\lambda$ is the typical count per interval. The Poisson has one famous fingerprint: $$\text{mean} = \lambda = \text{variance}.$$ The spread equals the average. If a real count's mean and variance are roughly equal, that's a strong hint a Poisson fits.
• The interpretation. It answers "how many in one chunk of time?" when events are independent and arrive at a steady average rate.

The core lesson counted every M4.5+ quake — a busy ~17 a day. Here we raise the bar to the bigger M5.0+ quakes, which are scarcer, and again tally how many happened each day in June 2024.

Look at the Poisson fingerprint. The mean (~3.6 quakes/day) and the variance (~2.7) are close — not glued together as tightly as the core's near-perfect match, but clearly in the same ballpark, which is the practical signal that a Poisson is reasonable. Nothing forces a dataset's mean and variance to match, so when they land near each other it's a real hint a Poisson process is at work. And notice the reindex step did its job: some June days saw zero M5+ quakes, and those 0s are now counted, not silently dropped. Now overlay the Poisson PMF (using $\lambda$ = the observed mean) on the histogram of daily counts.

The interpretation — and the contrast. With only 30 days the histogram is bumpy, but the red Poisson curve threads through it. The big story is the shape: at this low rate $\lambda \approx 3.6$, the distribution is visibly lopsided — it leans left, with a real chance of a totally quiet 0-quake day and a long thin tail toward busy days. Compare the core's $\lambda \approx 17$: that one was a fat, nearly symmetric mound far from zero. Same Poisson formula, but a rare rate makes it discrete and skewed, while a high rate makes it smooth and bell-like. (This is "this month's M5+ data"; a different month or cutoff would have its own $\lambda$.)

5) The bridge: Binomial becomes Poisson¶

These two models look unrelated, but they're secretly the same idea. Imagine slicing a day into a huge number $n$ of tiny instants, each a Bernoulli trial with a tiny chance $p$ of a big quake. That's a Binomial — but with $n$ enormous and $p$ minuscule. As $n \to \infty$ and $p \to 0$ while the product $\lambda = np$ stays fixed and moderate, the Binomial PMF turns into the Poisson PMF.

• The math. $\text{Binomial}(n, p) \to \text{Poisson}(\lambda)$ as $n\to\infty$, $p\to 0$, with $np = \lambda$.
• The meaning. Poisson is the "many trials, each rare" limit of the Binomial — exactly why it models rare events with no obvious $n$.

Let's overlay $\text{Binomial}(1000, 0.0036)$ on $\text{Poisson}(3.6)$ — matching this lesson's rarer quake rate — and watch them nearly coincide. (Note this is the rare-event $p$, the opposite end from the big survival $p$ of Part A — the bridge only works when $p$ is tiny.)

One sentence of meaning. When $n$ is large and $p$ is small with $np$ moderate, "successes out of $n$" and "rare events per interval" are the same distribution — so the Poisson is just the Binomial's natural shortcut for rare things, like a big quake on any given day.

Now you try¶

Predict first, then change one knob and re-run:

1. Crank up the success rate. In the section-3 simulation, the survival $p \approx 0.39$ already pushes the mound near the middle. Imagine a group where survival were $p = 0.74$ (the women-only rate from section 1): force it by replacing p with 0.74 in that cell. Which way does the whole histogram slide, and does it bunch up against the right wall (20) the way the births bunched against the left?

2. Lower the magnitude bar back to the core's. In section 4, change >= 5.0 to >= 4.5 and re-run sections 4. The rate $\lambda$ jumps to ~17, and the daily histogram should turn from lopsided into a near-symmetric mound — the core lesson's picture, recovered from the same file.

3. Break the Poisson bridge. In section 5, change n_big, p_small to 100, 0.036 (still $np = 3.6$, but now $p$ is ten times bigger). Does the Binomial still hug the Poisson, or does the gap grow? What does that tell you about how small $p$ has to be for the shortcut to work?

What you learned¶

• Bernoulli trial: one yes/no event with success probability $p$ — here, whether a passenger survived. The Survived column is a stack of them.
• Binomial $X\sim\text{Binomial}(n,p)$ counts successes in $n$ independent trials; $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$, with mean $=np$ and variance $=np(1-p)$ — confirmed by simulating random groups of 20 passengers. A bigger $p$ ($\approx 0.39$ vs. the births' $\approx 0.08$) shifts the whole distribution right and makes it more symmetric.
• Poisson $X\sim\text{Poisson}(\lambda)$ counts rare events per interval; $P(X=k)=e^{-\lambda}\lambda^k/k!$, with the fingerprint mean $\approx$ variance $=\lambda$ — seen in daily M5+ quake counts ($\lambda \approx 3.6$). A rarer rate makes the shape lopsided and choppy, versus the core's near-symmetric $\lambda \approx 17$.
• Counting zero-event intervals matters: reindexing over the full date range kept the quiet, zero-quake days in the tally.
• A model is believable when the simulation matches the formula and the data's mean/variance match the model's — not just because a textbook says so.
• Binomial $\to$ Poisson as $n\to\infty$, $p\to 0$ with $np=\lambda$: the two counting models are one idea, and the bridge needs the small-$p$ end, not the big survival $p$.

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