Is this similar to the p(A|B) and p(B|A) terms in Bayes Therom?

Question

Say there are a given number of seats at a table, and a given number are filled, and a given number are empty:

Percentage of seats empty = number of seats open / totals seats

Percentage of seats taken = number of seats taken / total seats

Percentage of seats empty + Percentage of seats open = 100%

Is this similar to the p(A|B) and p(B|A) terms in Bayes Therom?



Answer 1

First "proportion" is not always the same as "probability". It relies on the contents.

Probably, because we read $P(A|B)$ as "probability of A given B" you might think "given" is related to conditional probability.

However, conditional Probability $P(A|B)$ means "Probability that event A happens under the condition that event B happened".

First, the proportion of the empty seats means what Probability? Second, "Given number" means "a specific number". It doesn't mean "under the condition". Not related to the conditional probability at all.

Answer 2

No, it is just the principle of complements.   The sum of the measures of a set and its complement equals the measure of the total.

If $S$ is the set of seats open, and $T$ the set of all seats (so $S\subseteq T$), then the set of seats taken is the relative complement of $S$ over $T$ (ie $T{\smallsetminus}S$).

$$\lvert S\rvert+\lvert T{\smallsetminus}S\rvert=\lvert T\rvert$$

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