Bayes' theorem and evidence

Step 1: The four regions

Each region of $\mathrm{\Omega }$ is labelled with its number of cards: aces that are not red $2$, red aces $2$, red cards that are not aces $24$, and neither $24$. So there are $4$ aces and $26$ red cards.

Both conditional probabilities use the same overlap, the $2$ red aces.

Step 2: Given an ace

We learn that the card is an ace, so every card outside the aces is ruled out, and the $4$ aces become the whole. The overlap is half of it:

Step 3: Given red

Now we learn instead that the card is red, so every card outside red is ruled out, and the $26$ red cards become the whole. The same overlap is only a small part of it:

Step 4: Same overlap, different wholes

Both conditional probabilities divide the same overlap, $2$ cards, by the number of cards in the event we are given. The overlap is half of the small aces circle but only a small part of the large red circle.

So the conditional probability given the rarer event, ace, is the larger.