A mentally simpler, though slightly biased algorithm is for each item, randomly generate a uint64 (arbitrary bit size) and switch to the new item if and only if

A mentally simpler, though slightly biased algorithm is for each item, randomly generate a uint64 (arbitrary bit size) and switch to the new item if and only if the number generated is greater than or equal to all previously seen numbers. The end result is equivalent to randomly generating a number for each item and picking the item with the largest associated number.