There's an advert on the underground at the moment that says "Listerine users are more likely to talk to a stranger on the tube." I have a couple of issues with this advert: one ethical and one mathematical.

First the ethical. It sounds like they're suggesting that that's a good thing. But the guidance on talking to strangers on public transport is pretty clear, particularly if you're a man about to approach a woman: just ... don't.

Now that's out of the way, the mathematical issue is a little more subtle.

Like all good advertisers who want to stay on the right side of the ASA, and add a veneer of statistical credibility, they've included some small print, citing "research" they've done. They say that of 500 people surveyed, 125 would talk to a stranger on the tube. And of those 125 people, 80, or 64%, use Listerine. Compelling stuff.

Except that this fails to support their claim in any way.

The small print tells us that $P(\text{Listerine }|\text{ Annoying})=64\%$, the probability that you're a Listerine user, given that you talk to strangers, is 64%.
But their claim is that users are more likely to talk to strangers than non-users, or $P(\text{Annoying }|\text{ Listerine})>P(\text{Annoying }|\text{ Not Listerine})$.

In fact, based on the limited information given, Listerine users could be less likely to annoy innocent commuters than non-users. This is because we don't know anything about the mouthwash habits of those $500-125=375$ blessed people who leave their fellow passengers alone.

If all 375 saints are Listerine users then $$\begin{aligned} P(\text{Annoying }|\text{ Listerine})&=\frac{80}{375+80}=18\%\\ P(\text{Annoying }|\text{ Not Listerine})&=\frac{45}{45}=100\% \end{aligned}$$ and this mouthwash begins to look like an elixir of anti-harassment.

On the other hand, if none of those 375 swill then $$\begin{aligned} P(\text{Annoying }|\text{ Listerine})&=\frac{80}{80}=100\%\\ P(\text{Annoying }|\text{ Not Listerine})&=\frac{45}{375+45}=11\% \end{aligned}$$ which would support the advert's claim but... yikes.