I strongly disagree. One thing your math doesn't take in consideration is who is doing the action of putting the seat down. In fact, if we decide to put the seat down everytime, who is doing the action? 100% of the time, it's the man. How is that fair in any possible way?
In truth, if you consider the chance that the next user will either need the seat to be up or down, in a one man one woman household, the man should leave the seat down, and the woman should leave the seat up.
This is the math (not mine):
Let's assume that two people, a man and a woman, are living together.
Assume that men need to sit 33% of the time (I would say this is a high estimate, but I'll go with your numbers), and women need to sit 100% of the time.
Further assume that the likelihood of one party using the toilet twice in a row is 25% (that is, there's a 75% chance that the next person to use the toilet after the man is the woman, and vice versa.
Scenario 1: The woman uses the toilet. There is a 75% chance that the man will use the toilet next, and a (0.67 * 0.75) chance that the man will use the toilet and want the seat in the "up" position. This is 50.25%. If we use my estimate, which is that there's a 25% chance the man will need to sit, the number becomes 56.25%.
Scenario 2: The man uses the toilet. There is only a 25% he will use it again next, for a (0.67 * 0.25) = 16.75% chance that leaving it "up" will be appropriate for the next person using the toilet. -- 18.75% assuming 25% sitting.
Conclusions:
In a one woman, one man household:
-Women should leave the toilet with the seat "up". This will be convenient for the next toilet-user 56.25% of the time.
-Men should leave the toilet with the seat "down". This will be convenient for the next toilet-user 81.25% of the time.