With all of the probabilities of the joint events available, it is simple to construct our marginal probabilities (for example $P(\text{Yes to Pizza})$ or $P(\text{Yes to Beer})$) and it is straightforward to compute our conditional probabilites (for example $P(\text{Likes Pizza}|\text{Not liking Beer})$). Use the formulas directly to check that you can do that.

First, we will visually think about what the conditional probability really is. Suppose we want to condition on those who like Pizza. In terms of the Venn Diagram, this means we are no longer interested in the entire Venn Diagram, only on the part where Pizza is a 'Yes'. Since we only care about this area now, we want that the probabilities over liking Beer or not to add to one. The formula now makes sense, we have that $$ P[Beer=Yes|Pizza=Yes] = \frac{P[Beer=Yes,Pizza=Yes]}{P[Pizza=Yes]} $$ and $$ P[Beer=No|Pizza=Yes] = \frac{P[Beer=No,Pizza=Yes]}{P[Pizza=Yes]} $$ As in the main text we need to divide by $P[Pizza=Yes]$ so that these probabilities sum to one.

To actually use the formula, we need the marginal probabilities. This is of course easy in a problem such as this, we have that $$ P(\text{Likes Pizza}) = P(\text{Likes Pizza and Beer})+P(\text{Likes Pizza but not Beer})$$ which is equal to $0.2 + 0.5 = 0.7.$ We can easily read this from the Venn Diagram or read across the 'Yes' row for Pizza in the table.

Now plugging in the numbers from the Venn Diagram or table we have that $$ P[Beer=Yes|Pizza=Yes] = \frac{0.2}{0.7} = \frac{2}{7} $$ and $$ P[Beer=No|Pizza=Yes] = \frac{0.5}{0.7} = \frac{5}{7}. $$

Going through the same steps allows us to work out the reverse conditional probability, i.e. that $$ P[Pizza=Yes|Beer=Yes] = \frac{0.2}{0.4} = \frac{1}{2} $$ and $$ P[Pizza=No|Beer=Yes] = \frac{0.2}{0.4} = \frac{1}{2}. $$

Clearly we have that $P[Beer=Yes|Pizza=Yes] \ne P[Pizza=Yes|Beer=Yes]$, the ordering of the conditional distributions does matter. This is because they are completely different objects, they are probabilities of very different outcomes. One of the mistakes people make when casually thinking about probability is to not pay attention to how the thing we are conditioning on is correct relative to how we understand what we are doing.

This problem is so common it has plenty of names. In law it is called the prosecutors fallacy. The idea here is that from data we might understand from say DNA evidence and previous trials the $P(\text{DNA Evidence|Innocent})$ but in the trial, if there is evidence, we want the $P(\text{Innocent|DNA Evidence})$. The first of these probabilities might be very small, but that does not mean the reverse conditioning statement we care about is also small. In medicine it is known as "Zebras", which at least is a more interesting name. But it is the same problem, mistaking the conditional probability statement for its reverse. The reason for the name is that it is taught to medical students, when seeing a patient testing positive for a rare disease, that "If you see zebras, you hear hoofbeats but if you hear hoofbeats it probably is not a zebra". Medical students in the Serengeti probably have a different name for this mistake.

One additional name that is worth going through is what is known as the 'Base Rate Fallacy'. The base rate fallacy is mathematically similar to the issues we have been going through, but common enough that an additional example is useful. During covid there was a lot of misinformation. One that was prevalent was that half of new cases were amongst the vaccinated, and half of new cases were amongst the unvaccinated, so vaccines had no effect (the probabilities are the same). This is a classic example of the base rate fallacy - not taking into account that most people are vaccinated. Suppose the following (made up for the purposes of this example) table was correct.