Logic Puzzles

2/3rds.

The easiest way to see why the answer is 2/3 is to imagine repeating the experiment many times.

Suppose you do it 800 times.

The two coins have four equally likely outcomes:

HH, HT, TH, TT.

So roughly:

200 times they're HH.
200 times they're HT.
200 times they're TH.
200 times they're TT.

Now throw away every experiment where you and your friend didn't both see Heads.

When the coins are HH, both cups contain Heads. So every one of those 200 experiments survives. You both necessarily see Heads.

When the coins are HT, there's only one Heads cup. You need to pick that cup, which happens half the time. Then your friend also needs to independently pick that same cup, which happens half the time.

So both see Heads only:

1/2 × 1/2 = 1/4 of the time.

Of the 200 HT experiments, only 50 survive.

Exactly the same thing happens with TH. Of those 200 experiments, 50 survive.

TT can never produce two observations of Heads, so 0 survive.

So after learning that you both saw Heads, the possibilities remaining are:

HH: 200 cases
HT: 50 cases
TH: 50 cases

That's 300 possible cases, and 200 of them have both coins Heads.

So:

200 / 300 = 2/3.

The unintuitive part is that seeing Heads twice doesn't mean you've probably seen two different coins. If there's only one Heads coin, you both have to randomly choose that exact same cup. That's relatively unlikely. If both coins are Heads, however, seeing Heads twice is guaranteed.

That's why your two observations provide evidence in favor of HH, raising its probability from the original 1/4 to 2/3.