I'll try again. This time without python.

You throw two coins 1000 times. Each time, you write down the result. On average, you will then get roughly 250 heads-heads, 250 tails-tails, 250 heads-tails and 250 tails-heads. I hope we all agree on this.

You then select one of those results you just wrote down, and tell your friend that at least one of the coins came up heads in that throw. What is the probability that the other coin was also heads?

Well, since one of the coins was heads, you can disregard all the tails-tails results. That means that you could have one of the 250 heads-tails results, or one of the 250 tails-heads results, or one of the 250 heads-heads results. Hence, the probabilty that it was heads-heads is only 1/3.

Also, I don't know why this is relevant, but to get 1/3 from powers of 2, you can take (1/2^2)1-1/2^2). Which is essentially what we are doing here.