getting an average of a count of a group sql impala - sql

Question

Given a table like;
+----+----------+
| id | modality |
+----+----------+
| 1 | CT |
| 1 | SC |
| 1 | MR |
| 1 | CT |
| 2 | SC |
| 3 | CT |
| 3 | CT |
| 3 | MR |
| 4 | MR |
| 4 | CT |
+----+----------+
Something like;
SELECT AVG(modality_count)
FROM (
SELECT id, COUNT(modality) AS modality_count
FROM TABLE
GROUP BY id
) a
Would get the average amount of modality per patient, and output something like:
+---------------------+
| AVG(modality_count) |
+---------------------+
| 1.4 |
+---------------------+
the following query:
SELECT CONCAT(CAST(COUNT(*) AS string), ' ' , modality) AS count_m
FROM TABLE
GROUP BY modality
would get the count of the number of rows for each modality, like:
+---+---------+
| | count_m |
+---+---------+
| 1 | 560 CT |
| 2 | 2 SC |
| 3 | 473 MR |
+---+---------+
Let's say there are 1035 rows, 726 unique ids, and each id is likely to have 1.4 scans (either CT, SC, MR). How would you get the average number of CT/ MT/ SC?
desired output something like:
+----------+---------+
| modality | AVG(id) |
+----------+---------+
| CT | 0.77 |
| SC | 0.003 |
| MR | 0.65 |
+----------+---------+
Thanks in advance