Getting percentages with HAVING and CASE in SQL

percentages_sql_having_case

I have a table of accounts and a table of transactions.	There's	a
one-to-many relationship between accounts and transactions;                                                                                                                                                      
transactions.account_id is a foreign key into accounts.	For all
accounts where < 50% of transactions are authorized, I want to
get the account id and that percentage:

SELECT a.id, SUM(t.authorized=1)/COUNT(*) AS percentage
FROM accounts a, transaction_info t
WHERE a.id=t.account_id
GROUP BY a.id
HAVING percentage < .5 ORDER BY percentage DESC;                                                                                                                                                                 

What if transaction_info.authorized weren't already a boolean?
Let's say it's a VARCHAR with possible
values "authorized", "unauthorized", "unknown":

SELECT a.id, SUM(CASE WHEN l.authorized="authorized" THEN 1 ELSE 0 END)/COUNT(*) AS percentage
FROM accounts a, transaction_info t
WHERE a.id=t.account_id
GROUP BY a.id
HAVING percentage < .5 ORDER BY percentage DESC;