This post explores why some harms take decades to become statistically visible while others reveal themselves quickly. Smoking-related cancer is modeled as a slow, multi-stage process where the relevant “dose” accumulates over years. Vaccine adverse events, if they exist at a fixed per-dose rate, should appear after relatively few doses in a large enough population. The point is not that all risks are equal, but that rare events require reasoning about both individual probability and population scale.
Context from Taleb’s Twitter exchange
Some risks, like cancer from smoking, require a long “dose” (many years) before an event can occur, making the probability of a quick event (like cancer after one day) astronomically small. For vaccines, the “dose” is just a few injections, so if a rare adverse event is possible, it will show up quickly and does not require billions of people to observe.
Taleb:
“You don’t get throat cancer ‘in a day’ (or a week) because the mean dose is pack‑years (around 15!), so the tail is far, totally unattainable—you need trillions of smokers. Vaccines under consideration have a mean dose of 2–4 injections, so 1 dose is enough for inference.”
Refer to Taleb’s original tweet for context:
https://x.com/nntaleb/status/1646836313511829507?lang=ar-x-fm
This page illustrates both the individual-level risk functions and the corresponding population-level sample sizes for smoking-induced cancer and rare vaccine adverse events, following Nassim Taleb’s insights and an additional section for Gambling.
1. Smoking → Cancer (Erlang/Gamma)
This section uses the Erlang (Gamma) distribution to model how long it takes for a smoking-induced cancer to appear in an individual. It then computes the population size required so that, on average, one cancer case is observed within specified time horizons.
Model & formula:
$$
\displaystyle
f(t;,k,\lambda) = \frac{\lambda^k,t^{,k-1}e^{-\lambda t}}{(k-1)!},
\quad
F(t)=P(T\le t)=1-\sum_{i=0}^{k-1}e^{-\lambda t}\frac{(\lambda t)^i}{i!}.
$$
- $k = 5$ mutational stages
- $\lambda = k/15$ per year (so mean $\approx 15$ years)
Below is a table showing:
- $F(t)$: the individual probability of developing cancer by time $t$.
- $N$: the population size required to have, on average, one case ($N = 1/F(t)$).
| Time horizon | $t$ (years) | Individual $F(t)$ | Smokers needed $N=1/F(t)$ |
|---|---|---|---|
| 1 day | $1/365$ | $5.3\times10^{-18}$ | $1.9\times10^{17}$ |
| 1 week | $7/365$ | $9.1\times10^{-14}$ | $1.1\times10^{13}$ |
| 1 month | $30/365$ | $1.3\times10^{-10}$ | $7.8\times10^{9}$ |
| 1 year | $1$ | $2.6\times10^{-5}$ | $3.8\times10^{4}$ |
| 10 years | $10$ | $0.24$ | $4.1$ |
| 20 years | $20$ | $0.77$ | $1.3$ |
| 100 years | $100$ | $\approx1$ | $\approx1$ |
Interactive smoking model
2. Vaccines → Adverse Events (Exponential/Poisson)
This section treats rare vaccine adverse events as a constant-rate Poisson process, using the exponential distribution to model individual waiting times. It also calculates how many individuals are needed to expect one event by a given number of doses.
Model & formula:
For a constant hazard $\lambda$ per dose, the waiting time to the first event is exponential:
$$
f(t) = \lambda e^{-\lambda t},
\quad
F(t)=1-e^{-\lambda t}.
$$
And for $N$ people, the count of events after $t$ doses is Poisson with mean $N\lambda t$.
Assuming $\lambda=10^{-6}$ per dose (1 event per million doses), we have:
| Doses $t$ | Individual $F(t)=1-e^{-\lambda t}$ | People needed $N=1/F(t)$ |
|---|---|---|
| 1 | $1.0\times10^{-6}$ | $1,000,000$ |
| 2 | $2.0\times10^{-6}$ | $500,000$ |
| 5 | $5.0\times10^{-6}$ | $200,000$ |
| 10 | $1.0\times10^{-5}$ | $100,000$ |
Interactive vaccine adverse-event model
3. Gambling → 8-win Streak (Geometric)
This section models the chance of an 8-win streak in Bernoulli trials using the geometric distribution for waiting-blocks. It presents both the probability for an individual and the expected number of streaks in a large population.
Model & formula:
$$
P(N=n) = (1 - p^k)^{n-1} p^k
$$
where p is the per-play win probability (0.5) and k = 8.
Individual-level metrics:
| Metric | Value |
|---|---|
| p | 0.5 |
| k | 8 |
| Chance of an 8-win streak | $p^k = 0.5^8 = 0.0039$ $(\approx 0.39%)$ |
| Gamblers needed for one streak | $1 / p^k = 1 / 0.0039 \approx 256$ |
Note: “Gamblers needed for one streak” refers to the expected number of independent gamblers required so that, on average, one gambler achieves an 8-win streak.
Population-level metrics (N = 8×10⁹ gamblers):
| Metric | Calculation | Value |
|---|---|---|
| Expected streaks per day | $N \times p^k$ | $8 \times 10^9 \times 0.0039 \approx 3.12 \times 10^7$ |
| Average chance per gambler | $\frac{N p^k}{N} = p^k$ | 0.39% |
Even though each gambler’s chance is only $\approx 0.39%$, with 8 billion players you’d see around 31 million 8-win streaks per day.
Interactive gambling streak model
Gambling, astrology, and base rates
The same arithmetic explains why a fortune-teller, astrologer, or prediction account with billions of followers will appear uncannily accurate to many people. If a vague prediction has even a small chance of seeming true for any one person, then a planet-scale audience guarantees a large number of apparent “hits” by chance alone. The surprising part is not that some predictions come true; it would be surprising if none did.
Using the same $0.39%$ chance as the 8-win streak example:
| People receiving prediction $N$ | Chance prediction seems true | Expected chance hits $N \times 0.0039$ |
|---|---|---|
| $1,000,000$ | 0.39% | $\approx 3,900$ |
| $100,000,000$ | 0.39% | $\approx 390,000$ |
| $1,000,000,000$ | 0.39% | $\approx 3,900,000$ |
| $8,000,000,000$ | 0.39% | $\approx 31,200,000$ |