This is prompted by my attempts to work on this question. Let $H \subset G \subseteq S_d$ be transitive permutation groups. Recall that an element of $S_d$ is called a derangement if it has no fixed points.

Is the proportion of derangements in $H$ always greater than in $G$?

If $H$ doesn't have to be transitive, then the answer is "no"; just let $H$ be trivial. But a quick sampling of examples with $G$ and $H$ both transitive doesn't turn up any counterexamples.

UPDATE Never mind. $A_4$ inside $S_4$, the probability of a derangement goes down from $3/8$ to $1/4$.

This is prompted by my attempts to work on this question. Let $H \subset G \subseteq S_d$ be transitive permutation groups. Is the probability Recall that a random an element of $H$ has S_d$ is called a derangement if it has no fixed point points.

Is the proportion of derangements in $H$ always greater than or equal to the analogous probability for in $G$?

It is easy

If $H$ doesn't have to see that the expected number of fixed points increasesbe transitive, since this is equal to then the number of orbits of answer is "no"; just let $H$ (resp $G$) on $[d]$. Similar arguments show that the expected value be trivial. But a quick sampling of examples with $\binom{\# \mbox{fixed points}}{k}$ increases for every G$ and $k$. But it isn't obvious to me how to use that fact to prove that the probability of having a fixed point increasesH$ both transitive doesn't turn up any counterexamples.

This is prompted by my attempts to work on this question. Let $H \subset G \subseteq S_d$ be permutation groups. Is the probability that a random element of $H$ has a fixed point always greater than or equal to the analogous probability for $G$?

It is easy to see that the expected number of fixed points increases, since this is equal to the number of orbits of $H$ (resp $G$) on $[d]$. Similar arguments show that the expected value of $\binom{\# \mbox{fixed points}}{k}$ increases for every $k$. But it isn't obvious to me how to use that fact to prove that the probability of having a fixed point increases.