[pdurrant]

Originally Posted by obs20

Spoiler:

Okay what happens if the last 2 picked are white beans. The they are thrown away and 1 black bean is put back in the pot. Then you have a single black bean left.
Wait....
The last two were white that means the last was white. I get it, I get it

No, it's not a trick question. It's really asking what the colour is of the last bean in the pot after doing the described actions 224 (= 150+75-1) times. You have to stop then, because there's only one bean left in the pot, and the rules say to take out two beans.

Answer:

Spoiler:

The last bean in the pot is always white.

Black beans are removed from the pot one at a time. (Either one black and one white are removed, and only the white one put back, or two black beans are removed and one of them is put back)
White beans are only removed in pairs. (White beans are only discarded when a pair of white beans is removed.)

We start with an odd number of white beans. White beans are only removed in pairs. The number of beans in the pot goes down by one every time.

Since the number of beans goes down by one every time, eventually there must be only one bean in the pot. Since white beans are only removed in pairs, we started with an odd number, and white beans are never added to the pot, the last bean in the pot must be white. There's no point in considering the case where the last two beans in the pot are white — that cannot happen. There are never an even number of beans in the pot.