Smoking, Vaccines & Gambling

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 ($N = 1/F(t)$).

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:

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:

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):

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.