# Probability boxes (precious vs ordinary stones)

We play in a television game. We have 8 identical and indistinguishable boxes, each of which has 2 pebbles. Any pebble can be precious or not. We pick a box and the host, without looking at it, pulls out one of the pebbles. It turns out to be precious. The host then states that we have exactly 50% chance and the second pebble in this box is precious.

We assume that the host knows the distribution of the pebbles in the boxes - how many boxes the precious stones are 2, how many 1 and how many are 0. However, the host does not know what the pebbles are in a particular box because he does not distinguish the boxes himself.

If we know that initially the number of precious stones is not less than the number of precious, which of the following statements should be true?

A) If we ask for a change of box, we will have a better chance of the next drawn stone being precious.

B) If we ask for a change of box, we will not change our chances that the next downloaded stone will be precious.

C) If we ask for a change of box, we will have less chance of the next drawn stone being precious.

D) Initially, there were just 2 boxes with two precious stones in the game.

E) At first half the boxes had one precious and one precious stone.

F) Initially, there were an equal number of precious and precious stones in the game.

G) Initially the number of boxes with 2 precious stones is equal to the number of boxes with 2 precious stones.

H) The host is confused. There is no chance for a second precious stone in our box to be exactly 50% because there are an odd number of pebbles in the game.