on a/the toss of a/the coin
1. Literally, based on the outcome of a coin toss (i.e., the 50% chance that it will be either heads or tails). The team who starts off with the ball is always decided on the toss of a coin. We both wanted the last slice of pizza, so we're letting the outcome be determined on a toss of a coin.
2. By extension, purely at random; relying entirely on chance. New restaurants come and go in this area, their fates essentially decided on the toss of a coin. The economy is such an unpredictable beast that it seems to change on a toss of a coin.