Regarding the brain teaser, I’d say that it is better to stick with ‘up’ because if you start switching, you are unlikely (unless you make a point of it) to equally select ‘up’ and ‘down’. If you make an unequal selection of up/down, then you’ll be likely to miss more than 50% of the time as a result. (I.e., if you choose ‘up’ 60% of the time and ‘down’ 40%, and there is really a 50/50 split in the results, you will do more poorly than someone who just stuck with ‘up’ on 100% of their choices.)
OK, now I need to argue with myself: regardless the overall distribution of choices, each individual choice has a 50/50 chance of matching the result for each particular trial. So I’m not sure I see why it matters to stick with one result over randomly switching.