There is some empirical evidence that the probability that a permutation group of a given degree is soluble increases with the number of prime divisors of the degree. (Primes are counted with multiplicity so, for instance, $12$ has $3$ prime divisors.) I've tabulated the average proportion of soluble groups of degree $d$, for $5\leq d\leq 47$ (groups of degree less than $5$ are all soluble) according to the number $\Omega(d)$ of prime divisors of $d$.
| $\Omega(d)$ | Average Proportion |
|---|---|
| $1$ | $0.66623$ |
| $2$ | $0.65839$ |
| $3$ | $0.78775$ |
| $4$ | $0.95977$ |
| $5$ | $0.99962$ |
This suggests that a transitive group is more likely to be soluble if its degree has more prime divisors. In fact, it appears from this data that the likelihood that a transitive group of a given degree is soluble tends to $1$ as the number of prime divisors of that degree tends to $\infty$. (Granted, the only available degree with $5$ prime divisors is $32$; it would be nice to know the proportion for degree $48$.)
Question. Is this known to be true in general and, if so, is it understood why this would be the case?
Added (2022-05-08): With the data kindly provided by @DerekHolt, the average proportion for $\Omega(d)=$ changes very slightly to be $0.99956$.
Added (2022-05-15): Below is a plot of the proportions of soluble groups of each prime degree $p$ in the range $5\leq p\leq 4093$. A comment of @verret prompted me to take a second look at this data.
Added (2022-05-16): Plotting the average proportions of soluble groups of prime degrees $p$ in the range $5\leq p\leq 4093$, as suggested by @verret, yields the following plot. This seems to mostly confirm the behaviour predicted by @verret, though it is not clear that the limit is $1$. With the available data, the limit appears to approach something around $0.85$; perhaps the limit would reach $1$ eventually, but it is not clear from the data at hand.
