On September 23 at the United Nations, Donald Trump painted a picture of Europe in chaos, migrants overrunning the continent. His words were classic Trump — dramatic, sweeping, and backed by a statistic that sounded shocking.
But when you move from the headline to the math, the picture changes. Trump’s “72%” line about Switzerland’s prisons sounds like proof of a crisis. In reality, it’s a textbook case of how leaving out base rates distorts the truth — and why Bayesian thinking is essential for making sense of numbers in politics.
“Switzerland, beautiful Switzerland, 72% of the people in prisons are from outside of Switzerland.” — Trump at the U.N.
The number is technically correct — but by skipping the context, it becomes deeply misleading.
What Trump’s figure really means is: if you’re in prison in Switzerland, there’s a 72% chance you’re not Swiss. In probability terms:
\[P(\text{non-Swiss} \mid \text{prisoner}) = 72\%.\]But what many listeners heard is something very different: if you meet a non-Swiss person in Switzerland, there’s a high chance they’re in prison. That would be:
\[P(\text{prisoner} \mid \text{non-Swiss}).\]Those two probabilities are not the same thing. This is where Bayes’ theorem helps us see the difference:
\[P(\text{prisoner} \mid \text{non-Swiss}) = \frac{P(\text{non-Swiss} \mid \text{prisoner}) \times P(\text{prisoner})}{P(\text{non-Swiss})}.\]Trump gave the 72% piece — but left out the base rates that actually define the story.
Now apply Bayes with the actual values:
\[P(\text{prisoner} \mid \text{non-Swiss}) = \frac{0.72 \times \tfrac{7{,}000}{9{,}000{,}000}}{0.27} \approx 0.00208 = 0.21\%.\]So the probability that a non-Swiss resident is in prison is about 0.21% — roughly 1 in 500.
For comparison, for Swiss citizens:
\[P(\text{prisoner} \mid \text{Swiss}) = \frac{0.28 \times \tfrac{7{,}000}{9{,}000{,}000}}{0.73} \approx 0.00029 = 0.03\%.\]So the probability that a Swiss resident is in prison is about 0.03% — roughly 1 in 3,000.
Yes, non-Swiss residents are imprisoned at a higher rate — roughly 7× higher. But in absolute terms, more than 99.7% of non-Swiss residents are not in prison.
This is the sleight of hand. Trump takes a number that’s technically correct, strips away the base rates, and lets the audience imagine collapse. It’s like saying “most NBA players are tall” and then implying “most tall people are NBA players.” One is true. The other is absurd.
When he adds flourishes like “Europe is in serious trouble… London wants Sharia law” (rev.com), the pattern is clear: take a scary number, drop the base rates, let the audience imagine the worst.
Headlines can shock. But Bayes’ rule makes us slow down, ask the right question, and see what the numbers really mean.
Trump’s “72%” sounds like a crisis. With the math in place, it’s just another example of how statistics, stripped of context, can mislead.