For anyone looking for further explanation:
This is a conditional probability question, meaning we must update the probabilities of having chosen a certain box , given that we have already chosen a gold coin.
Note that the gold coins are not evenly distributed among all the boxes. For an example that should intuitively feel correct lets look at the extremes here. It should feel correct that if we drew a gold coin it is more likely we are drawing out of the box with ALL gold coins than the box containing only 1 gold coin. Upon knowing this new information ( drawing a gold coin ) our probabilities shift from the evenly distributed chance of picking a certain box of 1/4.
For a really obvious case imagine simply 2 boxes each with a million coins: one contains 1 gold coin the rest sliver, the other is exactly the opposite. Now if you have done this game again and you have a drawn a gold coin it should be apparent that you are
way more likely to have chosen the box with 999,999 gold coins, as opposed to drawing the single gold out of the box containing 999,999 silver coins., and that is why we update the probalites from 1/2 for this case.
So we are adding up the probabilities of being in a certain box ( given a first coin gold )
and then drawing a silver coin from that box.
All Gold
P( Box 1 ) = 4/10 ( box 1 contains 4 out of 10 total gold coins )
P( S | Box 1) = 0 ( there are no silver coins in Box 1 )
3G,1S
P( Box 2 ) = 3/10 ( box 2 contains 3 out of 10 total gold coins )
P( S | Box 2) = 1/3 ( if we have chosen box 2 there will be 1 silver coin and 2 gold coins remaining in the box )
2G,2S
P( Box 3 ) = 2/10 ( box 3 contains 2 out of 10 total gold coins )
P( S | Box 3) = 2/3 ( if we have chosen box 3 there will be 2 silver coin and 1 gold coin remaining in the box )
1G,3S
P( Box 4 ) = 1/10 ( box 4 contains 1 out of 10 total gold coins )
P( S | Box 3) = 1 ( if we have chosen box 4 there will be 3 silver coins remaining in the box - as we have removed the only gold )
Thus;
P(S | first draw G ) = 4/10 ‧ 0 + 3/10 ‧ 1/3 + 2/10 ‧ 2/3 + 1/10 ‧ 1 = 1/3
If anyone wants clarification on any part please let me know.
Hint:
Reveal Hidden ContentThis might all might be perfectly relevant to solve the Joe Shmo Show!