# If I have 200 coins with 100 being counterfeit placed in 2 boxes, what are the chances I'll get a real one?

I have 200 gold coins, but half of them are counterfeit (but no one can tell) and 2 boxes.

A)If someone randomly puts 100 coins in each box and I blindly pull one coin from a box of my choosing, what are the chances I get a real coin?

B)If I’m allowed to dictacte how the coins go in the boxes, what are my chances then? (doesn’t have to be 100 in each box, and still pulling blindy so I can’t see the coins inside of the box)

B2) If another person chooses the arrangement, how bad can my chances be then?

C) How would your answer to B change if there were 3 boxes instead of 2?

I’m not sure for A but isn’t it a 50/50 chance either way? and for B it’s still 50/50 no matter, and for C if you do 50 real in 2 boxes and 100 fake in the other, you have a 2/3 chance?

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A) should be 50/50 no matter what. You can prove it to yourself if you really want:

P(first box) = 1/2

P(second box) = 1/2

P(real coin in first box) =x/100

P(real coin in second box) = (100-x)/100

1/2 * x/100 + 1/2 * (100-x) / 100 = x + (100-x) / 200 = 100/200 = 1/2

B) I assume it means I can put the coins however I want? So let’s put 1 real coin in the first box and the rest of the coins in the second box.

1/2 * 1 + 1/2 * 99/199 = 0.7487

NICE!

B2) Now I’m pretty sure I’ve maximized our chances by making one box only contain one real coin. So another person could minimize our chance by rigging one box with one fake coin:

0 + 1/2 * 100/199 = 0.2513 (Which is pretty much 1-B)

C) If there were three boxes I would rig them again with two having one real coin and the rest of the coins in the third box:

1/3 + 1/3 + 1/3 * (98/198) = 0.8316