If there are 120 civilizations broadcasting radio signals in the Milky Way on average how many stars would we have to search before expecing to hear a signal assuming 500 billion stars in the galaxy?

Answer 1

The simple answer would be to divide 500 billion by 120, which gives an answer of slightly over four billion. The reality might be different, however. Just because a civilization is broadcasting radio signals does not mean that we necessarily can detect those signals. The Milky Way galaxy is 200,000 light years across on its long axis (80,000 on its shorter axis) and there are some stars which are so far away that it would take tens of thousands, even a hundred thousand years for a radio message to reach the Earth, and the hypothetical broadcasting civilization might not have been broadcasting for that long. And even if they were broadcasting long enough, the signal strength might not be enough to allow us to separate it from the background noise. And we might not be listening on the same frequency on which they are broadcasting. Those are just the most immediate complications.

Answer 2

If there are 120 civilizations broadcasting signals on average, the probability of finding one at a particular star is 120/500 billion. To determine how many stars we need to search, we would take the inverse of this probability: 500 billion/120 ≈ 4.17 billion stars on average before expecting to hear a signal.