We are in a TV game. We have 8 same looking boxes, each of which has 2 pebbles. Each pebble could be either precious or not precious. We choose a box and the host, without looking inside the box, pulls out a pebble. It comes out to be precious. The TV host then declares that we have exactly 50% chance that the 2nd pebble inside the box is precious, too.
If we know that, in the beginning of the game, the count of precious pebbles is no less than the count of not precious pebbles, which of the statements MUST be true?
We assume that the TV host knows the pebbles distribution in the boxes - in what amount of boxes the precious pebbles are 0,1 or 2. The TV host, however, doesn't know the kind of pebbles in each specific box, as he himself doesn't differ the boxes from one another.
1. If we swapped the boxes, we would have higher chance of picking up precious pebble next.
2. If we swapped the boxes, we would have same chance of picking up precious pebble next.
3. If we swapped the boxes, we would have lower chance of picking up precious pebble next.
4. In the beggining of the game, there've been exactly 2 boxes with 2 precious pebbles each.
5. In the beginning of the game, half the boxes have had 1 precious and 1 not precious pebble.
6. In the beginning of the game, the precious and not precious pebbles have been equal in count.
7. In the beginning of the game, the amount of boxes with 2 precious pebbles is equal to the amount of boxes with 2 not precious pebbles.
8. The TV host has gone wrong. There's no way that there's 50% chance for 2nd precious pebble in our box, as there is unequal amount of pebbles left in the game.
9. None of the above must be true.