If $P(x \mid y)$ is high, then does this imply that $P(y \mid x)$ is also high? If $P(x \mid y)$ is high, for example, above 0.5, does it imply that $P(y \mid x)$ is also high (above $0.5$)? If yes/no, then why? What if I know that they are dependent on each other? For what type of events may this relation hold?
 A: $P(x|y)$ and $P(y|x)$ are different things and do not have to be related. Say that you have an “algorithm” that predicts the weather. It is very simple, it always predicts “it is going to rain”. So P(predicted rain|rain) = 1, in all cases it rained, it also predicted rain. On another hand, say that in your area it rains only 5% of the time, so P(rain|predicted rain) = 0.05, since only in 5% of cases when it predicted rain (always), it rained.
Think of it geometrically. Say that you are throwing stones into a box split into four quadrants while being blindfolded (i.e. at random). If I tell you that you threw the stone into the quadrant within the $x=1$ column, the conditional probability that it was $y=1$ (you're still blindfolded!), $p(y=1|x=1)=1/2$ would be the area of $y=1,x=1$ quadrant, divided by the area of the $x=1$ column. For $p(x=1|y=1)=1/2$, the calculation would be done row-wise.
$$
\begin{array}{c|c|c|} 
               & x=0 &  x=1 \\ 
\hline
 y=0  &  &  \\ 
\hline 
 y=1  &  &  \\ 
\hline
\end{array}
$$
Now imagine that you expand the area of the $y=1$ row twice. The marginal $p(y=1)=2/3$ and conditional probabilities $p(y=1|x=1)=2/3$ have changed. However, if you look at the relative areas from the $x$'s point of view, the relative areas $p(x=1|y=1)=1/2$ are still the same.
$$
\begin{array}{c|c|c|} 
               & x=0 &  x=1 \\ 
\hline
 y=0  &  &  \\ 
\hline 
 y=1\\\phantom{y=1}  &  &  \\
\hline
\end{array}
$$
The same can be done if you expanded the area of any of the columns instead, the result would be the opposite.
If you changed the areas on both of the dimensions, both the conditional probabilities would change. Play around with it (in your head, using pen and pencil, or using a computer) and check how do the relative areas change. If you expand the row together with the column, say making $x=1$ twice as big and $y=1$ twice as big, would make both conditional probabilities equal to $2/3$.
$$
\begin{array}{c|c|c|} 
               & x=0 & x=1\phantom{x=1} \\ 
\hline
 y=0  &  &  \\ 
\hline 
 y=1\\\phantom{y=1}  &  &  \\
\hline
\end{array}
$$
But you could also do the scaling in an opposite way, expand $x=0$ and $y=1$, in such case $p(x=1|y=1)=1/3$ and $p(y=1|x=1)=2/3$.
$$
\begin{array}{c|c|c|} 
               & x=0\phantom{x=0} & x=1 \\ 
\hline
 y=0  &  &  \\ 
\hline 
 y=1\\\phantom{y=1}  &  &  \\
\hline
\end{array}
$$
A: Let's review the conditional probability of both $P(x|y)$ and $P(y|x)$:
$$P(x|y) = \frac{P(x \cap y)}{P(y)}$$
and
$$P(y|x) = \frac{P(x \cap y)}{P(x)}$$
The common thing in both is their joint probability. So, their discrepancy depends on $P(x)$ and $P(y)$. If these two probabilities are equal or close to each other then according to formulas, the conditional probabilities are close to each other. This also depends on the degree of dependence between $x$ and $y$.
For example consider the events of $T:\text{Thunder}$ and $B:\text{Black clouds}$. They often come together and they strongly depend on each others, so approximately you may write:
$$ P(T) \approx P(B) \approx P(T \cap B)$$
Then the conditional probabilities are equal:
$$P(B | T) \approx P(T | B) \approx 1 $$
Now consider the example of $H:\text{Being hanged}$ and $D:\text{Death}$, which one is the cause of the other one but not vice versa. We can safely assume whenever one is hanged, they die, and it's the only usage of hanging. So:
$$ P(H) \approx P(D \cap H)$$
Therefore:
$$P(D | H) = \frac{P(H \cap D)}{P(H)} \approx 1$$
However, death could be due to many other reasons than being hanged and $$P(D) \gg P(D \cap H)$$
Therefore:
$$P(H | D) = \frac{P(H \cap D)}{P(D)} \approx 0$$
A: I like this example:
$$ \DeclareMathOperator{\P}{\mathbb{P}}
\P(\text{Dead} \mid \text{Was hanged}) \qquad \text{is high}\\
\P(\text{Was hanged} \mid \text{Dead}) \qquad \text{is low}
$$
and that should answer your question.
